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High School Math : Trigonometry

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Example Questions

Example Question #43 : Understanding Radians And Conversions

How many radians are in $208^\circ$ ?

Possible Answers:

$1.39\pi$

$1.15\pi$

$1.07\pi$

$1.67\pi$

$1.11\pi$

Correct answer:

$1.15\pi$

Explanation:

To convert degrees to radians, set up a ratio. The ratio of degrees to radians is $\frac{180^\circ}{\pi}$ .

$\frac{208^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply.

$208^\circ*\pi=180^\circ*x$

$\frac{208^\circ}{180^\circ}*\pi=x$

$1.15\pi=x$

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Example Question #54 : The Unit Circle And Radians

How many radians are in $313^\circ$ ?

Possible Answers:

$2.11\pi$

$1.74\pi$

$0.56\pi$

$3.48\pi$

$1.05\pi$

Correct answer:

$1.74\pi$

Explanation:

To solve this, use a proportion. The ratio of degrees to radians is $\frac{180^\circ}{\pi}$ .

$\frac{313^\circ}{x}=\frac{180^\circ }{\pi}$

Cross multiply:

$313^\circ \pi=180^\circ x$

$\frac{313^\circ}{180^\circ }\pi=x$

$1.74\pi=x$

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Example Question #81 : Trigonometry

In the unit circle, what is the angle in radians that corresponds to the point (0, -1)?

Possible Answers:

$2\pi$

$\frac{\pi }{2}$

$\pi$

$\frac{3\pi }{2}$

Correct answer:

$\frac{3\pi }{2}$

Explanation:

On the unit circle, (0,-1) is the point that falls between the third and fourth quadrant. This corresponds to $\frac{3\pi }{2}$ .

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Example Question #82 : Trigonometry

What is the reference angle for $\frac{16\pi}{3}$ ?

Possible Answers:

$\frac{\pi}{3}$

$\frac{3\pi}{4}$

$\frac{4\pi}{3}$

$\frac{\pi}{6}$

Correct answer:

$\frac{4\pi}{3}$

Explanation:

To find the reference angle, subtract $2\pi$ (1 trip around the unit circle) from the given angle until you reach an angle which is less than $2\pi$ .

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Example Question #1 : Triangles

Rt_triangle_letters In this figure, side X=20 , Y=13 , and $c=90^\circ$ . What is the value of angle ?

Possible Answers:

Undefined

$40.54^\circ$

$6.5^\circ$

$49.46^\circ$

$90^\circ$

Correct answer:

$49.46^\circ$

Explanation:

Since $c=90^\circ$ , we know we are working with a right triangle.

That means that $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$ .

In this problem, that would be:

$\cos(a)=\frac{\text{Y}}{\text{X}}$

Plug in our given values:

$\cos(a)=\frac{\text{13}}{\text{20}}$

$\cos(a)=0.65$

$a=\cos^{-1}(0.65)$

$a=49.46^\circ$

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Example Question #2 : Triangles

Let ABC be a right triangle with sides = 3 inches, = 4 inches, and = 5 inches. In degrees, what is the $\sin \Theta$ where $\Theta$ is the angle opposite of side ?

Possible Answers:

Correct answer:

Explanation:

3-4-5_triangle

We are looking for $\sin(\theta )$ . Remember the definition of sin in a right triangle is the length of the opposite side divided by the length of the hypotenuse.

So therefore, without figuring out $\Theta$ we can find

$\sin \Theta =\frac{3}{5}=.6$

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Example Question #1 : Trigonometric Functions And Graphs

Rt_triangle_letters

In this figure, if angle $c=90^\circ$ , side X=12 , and side Z=8 , what is the measure of angle ?

Possible Answers:

Undefined

$\frac{2}{3}^\circ$

$48.2^\circ$

$90^\circ$

$41.8^\circ$

Correct answer:

$41.8^\circ$

Explanation:

Since $c=90^\circ$ , we know we are working with a right triangle.

That means that $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$ .

In this problem, that would be:

$\sin(a)=\frac{\text{Z}}{\text{X}}$

Plug in our given values:

$\sin(a)=\frac{8}{12}$

$\sin(a)=\frac{2}{3}$

$a=\sin^{-1}(\frac{2}{3})$

$a=41.8^\circ$

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Example Question #3 : Triangles

Rt_triangle_letters

In this figure, X=12 , $Y=6\sqrt{3}$ , and Z=6 . What is the value of angle ?

Possible Answers:

Undefined

$45^\circ$

$60^\circ$

$30^\circ$

$90^\circ$

Correct answer:

$30^\circ$

Explanation:

Notice that these sides fit the pattern of a 30:60:90 right triangle: $x:x\sqrt{3}:2x$ .

In this case, x=6 .

Since angle is opposite , it must be $30^\circ$ .

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Example Question #4 : Triangles

A triangle has angles of $30^\circ:60^\circ:90^\circ$ . If the side opposite the $30^\circ$ angle is , what is the length of the side opposite $60^\circ$ ?

Possible Answers:

$14\sqrt{3}$

$\frac{7\sqrt{2}}{2}$

$7\sqrt{3}$

$7\sqrt{2}$

Correct answer:

$7\sqrt{3}$

Explanation:

The pattern for $30^\circ:60^\circ:90^\circ$ is that the sides will be $x:x\sqrt{3}:2x$ .

If the side opposite $30^\circ$ is , then the side opposite $60^\circ$ will be $7\sqrt{3}$ .

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Example Question #1 : Triangles

In $\Delta ABC$ , AB = 26 , BC = 32 , and AC = 23 . To the nearest tenth, what is $m \angle A$ ?

Possible Answers:

$171.3^{ \circ }$

$81.3 ^{\circ }\textrm{ or } 98.7^{ \circ }$

A triangle with these sidelengths cannot exist.

$81.3 ^{\circ }$

$98.7^{ \circ }$

Correct answer:

$81.3 ^{\circ }$

Explanation:

By the Triangle Inequality, this triangle can exist, since 23 + 26 > 32 .

By the Law of Cosines:

$\left ( BC\right )^{2} = \left ( AB\right )^{2} + \left ( AC\right )^{2} - 2 \cdot AB \cdot AC \cdot \cos m \angle A$

Substitute the sidelengths and solve for $\cos m \angle A$ :

$32^{2} = 26^{2} + 23^{2} - 2\cdot 26 \cdot 23 \cdot \cos m \angle A$

$1,024 =676 + 529 - 1,196 \cdot \cos m \angle A$

$1,024 =1,205 - 1,196 \cdot \cos m \angle A$

$-181 = - 1,196 \cdot \cos m \angle A$

$\cos m \angle A = 181 \div 1,196 \approx 0.1513$

$m \angle A \approx \cos^{-1} 0.1513 \approx 81.3 ^{\circ }$

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High School Math : Trigonometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #43 : Understanding Radians And Conversions

Example Question #54 : The Unit Circle And Radians

Example Question #81 : Trigonometry

Example Question #82 : Trigonometry

Example Question #1 : Triangles

Example Question #2 : Triangles

Example Question #1 : Trigonometric Functions And Graphs

Example Question #3 : Triangles

Example Question #4 : Triangles

Example Question #1 : Triangles

All High School Math Resources

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