This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a sine equalling negative one?

$\displaystyle y=\sin^{-1}(a)\Rightarrow \sin(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in negative one for its sine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle y=-1$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{3\pi}{2}$ the sine equals negative one. 

Thus,

$\displaystyle f(x)=\sin^{-1}(-1)\Rightarrow f(x)=\frac{3\pi}{2}$

To verify the solution simply find the sine of the angle theta.

$\displaystyle \sin\left(\frac{3\pi}{2}\right)=-1\rightarrow\sin^{-1}(-1)=\frac{3\pi}{2}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling zero?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in zero for its cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=0$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{\pi}{2}$ and $\displaystyle \frac{3\pi}{2}$ the cosine equals zero. 

Thus,

$\displaystyle f(x)=\cos^{-1}(1)\Rightarrow f(x)=\frac{3\pi}{2},\frac{\pi}{2}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{3\pi}{2} \right )=0\rightarrow\cos^{-1}(0)=\frac{3\pi}{2} \\ \\\cos\left(\frac{\pi}{2} \right )=0\rightarrow\cos^{-1}(0)=\frac{\pi}{2}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling one half?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=\frac{1}{2}$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{\pi}{3}$ and $\displaystyle \frac{5\pi}{3}$ the cosine equals the given value. 

Thus,

$\displaystyle f(x)=\cos^{-1}\left(\frac{1}{2} \right )\Rightarrow f(x)=\frac{5\pi}{3},\frac{\pi}{3}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{5\pi}{3} \right )=\frac{1}{2}\rightarrow\cos^{-1}\frac{1}{2}=\frac{5\pi}{3} \\ \\\cos\left(\frac{\pi}{3} \right )=\frac{1}{2}\rightarrow\cos^{-1}\frac{1}{2}=\frac{\pi}{3}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling negative one half?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=-\frac{1}{2}$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{2\pi}{3}$ and $\displaystyle \frac{4\pi}{3}$ the cosine equals the given value. 

Thus,

$\displaystyle f(x)=\cos^{-1}\left(-\frac{1}{2} \right )\Rightarrow f(x)=\frac{4\pi}{3},\frac{2\pi}{3}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{4\pi}{3} \right )=-\frac{1}{2}\rightarrow\cos^{-1}\left( -\frac{1}{2}\right)=\frac{5\pi}{3} \\ \\\cos\left(\frac{2\pi}{3} \right )=-\frac{1}{2}\rightarrow\cos^{-1}\left( -\frac{1}{2}\right)=\frac{2\pi}{3}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling negative one over the square root of two?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=-\frac{1}{\sqrt{2}}$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{3\pi}{4}$ and $\displaystyle \frac{5\pi}{4}$ the cosine equals the given value. 

Thus,

$\displaystyle f(x)=\cos^{-1}\left(-\frac{1}{\sqrt{2}} \right )\Rightarrow f(x)=\frac{3\pi}{4},\frac{5\pi}{4}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{3\pi}{4} \right )=-\frac{1}{\sqrt{2}}\rightarrow\cos^{-1}\left( -\frac{1}{\sqrt{2}}\right)=\frac{3\pi}{4} \\ \\\cos\left(\frac{5\pi}{4} \right )=-\frac{1}{\sqrt{2}}\rightarrow\cos^{-1}\left( -\frac{1}{\sqrt{2}}\right)=\frac{5\pi}{4}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling one over the square root of two?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=\frac{1}{\sqrt{2}}$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{\pi}{4}$ and $\displaystyle \frac{7\pi}{4}$ the cosine equals the given value. 

Thus,

$\displaystyle f(x)=\cos^{-1}\left(\frac{1}{\sqrt{2}} \right )\Rightarrow f(x)=\frac{\pi}{4},\frac{7\pi}{4}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{\pi}{4} \right )=\frac{1}{\sqrt{2}}\rightarrow\cos^{-1}\left( \frac{1}{\sqrt{2}}\right)=\frac{\pi}{4} \\ \\\cos\left(\frac{7\pi}{4} \right )=\frac{1}{\sqrt{2}}\rightarrow\cos^{-1}\left( \frac{1}{\sqrt{2}}\right)=\frac{7\pi}{4}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling the square root of three over two?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=\frac{\sqrt{3}}{2}$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{\pi}{6}$ and $\displaystyle \frac{11\pi}{6}$ the cosine equals the given value. 

Thus,

$\displaystyle f(x)=\cos^{-1}\left(\frac{\sqrt{3}}{2} \right )\Rightarrow f(x)=\frac{\pi}{6},\frac{11\pi}{6}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{\pi}{6} \right )=\frac{\sqrt{3}}{2}\rightarrow\cos^{-1}\frac{\sqrt{3}}{2}=\frac{\pi}{6} \\ \\\cos\left(\frac{11\pi}{6} \right )=\frac{\sqrt{3}}{2}\rightarrow\cos^{-1}\frac{\sqrt{3}}{2}=\frac{11\pi}{6}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a cosine equalling negative square root of three over two?

$\displaystyle y=\cos^{-1}(a)\Rightarrow \cos(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given cosine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle x=-\frac{\sqrt{3}}{2}$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{5\pi}{6}$ and $\displaystyle \frac{7\pi}{6}$ the cosine equals the given value. 

Thus,

$\displaystyle f(x)=\cos^{-1}\left(-\frac{\sqrt{3}}{2} \right )\Rightarrow f(x)=\frac{5\pi}{6},\frac{7\pi}{6}$

To verify the solution simply find the cosine of the angle theta.

$\displaystyle \\\cos\left(\frac{5\pi}{6} \right )=-\frac{\sqrt{3}}{2}\rightarrow\cos^{-1}\frac{-\sqrt{3}}{2}=\frac{5\pi}{6} \\ \\\cos\left(\frac{7\pi}{6} \right )=-\frac{\sqrt{3}}{2}\rightarrow\cos^{-1}\frac{-\sqrt{3}}{2}=\frac{7\pi}{6}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a sine equalling one?

$\displaystyle y=\sin^{-1}(a)\Rightarrow \sin(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given sine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle y=1$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \frac{\pi}{2}$ the sine equals the given value. 

Thus,

$\displaystyle f(x)=\sin^{-1}(1)\Rightarrow f(x)=\frac{\pi}{2}$

To verify the solution simply find the sine of the angle theta.

$\displaystyle \sin\left(\frac{\pi}{2}\right)=1\rightarrow\sin^{-1}(1)=\frac{\pi}{2}$

This question is testing ones ability to understand and identify inverses of trigonometric functions as they relate to the unit circle.

For the purpose of Common Core Standards, " Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed." concept (CCSS.MATH.CONTENT.HSF-TF.B.6). It is important to note that this standard is not directly tested on but is used for building a deeper understanding on trigonometric functions.

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 what the question is asking for.

Since there is a trigonometric function raised to the negative one power, this question is talking about the inverse of the function. In other words, which angle on the unit circle results in a sine equalling zero?

$\displaystyle y=\sin^{-1}(a)\Rightarrow \sin(\theta)=a$

Therefore, theta needs to be solved for.

Step 2: Draw and label the unit circle.

Step 3: Locate the angle that results in the given sine value.

Recall that 

$\displaystyle (x,y)=(\cos(\theta),\sin(\theta))$

therefore look for the $\displaystyle (x,y)$ that has $\displaystyle y=0$. Looking at the unit circle from step 2, it is seen that at angle $\displaystyle \pi$ and $\displaystyle 2\pi$ the sine equals the given value. 

Thus,

$\displaystyle f(x)=\sin^{-1}(0)\Rightarrow f(x)=\pi \text{ or }2\pi$

To verify the solution simply find the sine of the angle theta.

$\displaystyle \\\sin\left({\pi}\right)=0\rightarrow\sin^{-1}(0)=\pi\\ \sin(2\pi)=0\rightarrow \sin^{-1}(0)=2\pi$