Common Core: High School - Functions : Radians and Arc Length: CCSS.Math.Content.HSF-TF.A.1

Study concepts, example questions & explanations for Common Core: High School - Functions

varsity tutors app store varsity tutors android store

All Common Core: High School - Functions Resources

6 Diagnostic Tests 82 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Radians And Arc Length: Ccss.Math.Content.Hsf Tf.A.1

Convert \(\displaystyle 135^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{\pi}{2}\)

\(\displaystyle \frac{4\pi}{6}\)

\(\displaystyle \frac{3\pi}{4}\)

\(\displaystyle \frac{\pi}{4}\)

\(\displaystyle \pi\)

Correct answer:

\(\displaystyle \frac{3\pi}{4}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.Math.content.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=135^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{135^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{135^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{135\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{135\pi}{180} \\ \\ =\frac{45\cdot 3\pi}{45\cdot 4} \\ \\ =\frac{3\pi}{4}\)

Example Question #1 : Radians And Arc Length: Ccss.Math.Content.Hsf Tf.A.1

Convert \(\displaystyle 45^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{\pi}{4}\)

\(\displaystyle \frac{\pi}{2}\)

\(\displaystyle \frac{3\pi}{4}\)

\(\displaystyle \frac{2\pi}{4}\)

\(\displaystyle \frac{\pi}{6}\)

Correct answer:

\(\displaystyle \frac{\pi}{4}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=45^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{45^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{45^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{45\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{45\pi}{180} \\ \\ =\frac{45\cdot 1\pi}{45\cdot 4} \\ \\ =\frac{\pi}{4}\)

Example Question #1 : Trigonometric Functions

Convert \(\displaystyle 80^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{9\pi}{4}\)

\(\displaystyle \frac{8\pi}{9}\)

\(\displaystyle \frac{4\pi}{5}\)

\(\displaystyle \frac{3\pi}{9}\)

\(\displaystyle \frac{4\pi}{9}\)

Correct answer:

\(\displaystyle \frac{4\pi}{9}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=80^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{80^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{80^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{80\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{80\pi}{180} \\ \\ =\frac{20\cdot 4\pi}{20\cdot 9} \\ \\ =\frac{4\pi}{9}\)

Example Question #1 : Trigonometric Functions

Convert \(\displaystyle 320^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{9\pi}{16}\)

\(\displaystyle \frac{16\pi}{9}\)

\(\displaystyle \frac{16\pi}{8}\)

\(\displaystyle \frac{16\pi}{4}\)

\(\displaystyle \frac{9\pi}{6}\)

Correct answer:

\(\displaystyle \frac{16\pi}{9}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=320^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{320^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{320^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{320\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{320\pi}{180} \\ \\ =\frac{20\cdot 16\pi}{20\cdot 9} \\ \\ =\frac{16\pi}{9}\)

Example Question #3 : Trigonometric Functions

Convert \(\displaystyle 275^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{5\pi}{6}\)

\(\displaystyle \frac{45\pi}{36}\)

\(\displaystyle \frac{5\pi}{3}\)

\(\displaystyle \frac{55\pi}{36}\)

\(\displaystyle \frac{36\pi}{55}\)

Correct answer:

\(\displaystyle \frac{55\pi}{36}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=275^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{275^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{275^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{275\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{275\pi}{180} \\ \\ =\frac{5\cdot55\pi}{5\cdot 36} \\ \\ =\frac{55\pi}{36}\)

Example Question #4 : Trigonometric Functions

Convert \(\displaystyle 75^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{5\pi}{12}\)

\(\displaystyle \frac{5\pi}{20}\)

\(\displaystyle \frac{5\pi}{2}\)

\(\displaystyle \frac{5\pi}{10}\)

\(\displaystyle \frac{12\pi}{5}\)

Correct answer:

\(\displaystyle \frac{5\pi}{12}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=75^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{75^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{75^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{75\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{75\pi}{180} \\ \\ =\frac{15\cdot 5\pi}{15\cdot 12} \\ \\ =\frac{5\pi}{12}\)

Example Question #3 : Trigonometric Functions

Convert \(\displaystyle 112^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{8\pi}{5}\)

\(\displaystyle \frac{18\pi}{45}\)

\(\displaystyle \frac{28\pi}{45}\)

\(\displaystyle \frac{45\pi}{28}\)

\(\displaystyle \frac{24\pi}{40}\)

Correct answer:

\(\displaystyle \frac{28\pi}{45}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=112^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{112^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{112^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{112\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{112\pi}{180} \\ \\ =\frac{4\cdot 28\pi}{4\cdot 45} \\ \\ =\frac{28\pi}{45}\)

Example Question #3 : Radians And Arc Length: Ccss.Math.Content.Hsf Tf.A.1

Convert \(\displaystyle 12^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{\pi}{12}\)

\(\displaystyle \frac{\pi}{5}\)

\(\displaystyle \frac{2\pi}{15}\)

\(\displaystyle \frac{12\pi}{15}\)

\(\displaystyle \frac{\pi}{15}\)

Correct answer:

\(\displaystyle \frac{\pi}{15}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=12^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{12^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{12^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{12\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{12\pi}{180} \\ \\ =\frac{12\cdot 1\pi}{12\cdot 15} \\ \\ =\frac{\pi}{15}\)

Example Question #1 : Radians And Arc Length: Ccss.Math.Content.Hsf Tf.A.1

Convert \(\displaystyle 33^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{\pi}{6}\)

\(\displaystyle \frac{60\pi}{11}\)

\(\displaystyle \frac{11\pi}{60}\)

\(\displaystyle \frac{10\pi}{62}\)

\(\displaystyle \frac{12\pi}{61}\)

Correct answer:

\(\displaystyle \frac{11\pi}{60}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=33^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{33^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{33^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{33\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{33\pi}{180} \\ \\ =\frac{3\cdot 11\pi}{3\cdot 60} \\ \\ =\frac{11\pi}{60}\)

Example Question #1 : Trigonometric Functions

Convert \(\displaystyle 90^\circ\) to radians.

Possible Answers:

\(\displaystyle \frac{\pi}{6}\)

\(\displaystyle \frac{\pi}{3}\)

\(\displaystyle \frac{3\pi}{2}\)

\(\displaystyle \frac{\pi}{2}\)

\(\displaystyle \frac{\pi}{4}\)

Correct answer:

\(\displaystyle \frac{\pi}{2}\)

Explanation:

This question tests one's ability to understand the conversion between radians and degrees. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each \(\displaystyle (x,y)\) pair that lies on the circle can be found by creating a right triangle that has a base on the \(\displaystyle x\)-axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the \(\displaystyle x\)-axis.

For the purpose of Common Core Standards, understanding radian measure of an angle as the length of the arc on the unit circle subtended by the angle, falls within the Cluster A of extend the domain of trigonometric functions using the unit circle concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Identify the general formula for unit conversion between radians and degrees.

\(\displaystyle 360^\circ=2\pi \textup{ radians}\Rightarrow180^\circ=\pi\textup{ radians}\)

Therefore to convert from degrees to radian is,

\(\displaystyle \frac{x^\circ}{1}\times \frac{\pi}{180^\circ}=\textup{radian measurement}\)

Step 2: Identify the values given in the question.

\(\displaystyle x=\textup{degrees}=90^\circ\)

Step 3: Substitute the value from step 2 into the formula from step 1.

\(\displaystyle \\ \frac{x^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{90^\circ}{1}\times \frac{\pi}{180^\circ} \\ \\ =\frac{90^\circ\pi}{180^\circ}\)

The degree measurement in the numerator and denominator cancel out. 

\(\displaystyle \frac{90\pi}{180}\)

From here, simplify the fraction by finding common factors that exist in both the numerator and denominator.

\(\displaystyle \\ \frac{90\pi}{180} \\ \\ =\frac{90\cdot 1\pi}{90\cdot 2} \\ \\ =\frac{\pi}{2}\)

All Common Core: High School - Functions Resources

6 Diagnostic Tests 82 Practice Tests Question of the Day Flashcards Learn by Concept
Learning Tools by Varsity Tutors