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 half?

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 

therefore look for the  that has . Looking at the unit circle from step 2, it is seen that at angle  and  the sine equals the given value. 

Thus,

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

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

First subtract four from each side.

Divide both sides by two.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

.

Step 4: Answer the question.

Solving for  results in two possible values, 

.

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

Divide each side by two.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

.

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

First subtract four from both sides.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

Divide by two on both sides.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

Divide by three on both sides.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

Divide by three on both sides.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

Add one to both sides.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

First add four from each side.

Divide both sides by two.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

.

Step 4: Answer the question.

Solving for  results in two possible values, 

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

For the purpose of Common Core Standards, "Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context" concept (CCSS.MATH.CONTENT.HSF-TF.B.7). 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: Use algebraic operations to manipulate the function.

Subtract one from each side.

Step 2: To isolate theta, perform the inverse trigonometric operation.

Step 3: Use the unit circle to solve for theta.

Recall that 

Therefore, to have tangent equalling one, both sine and cosine must be equal to one another.

Therefore, 

Step 4: Answer the question.

Solving for  results in two possible values,