Find the exact values of the six trigonometric functions of θ

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Solve the following:  

Correct answer:

Explanation:

Rewrite  in terms of sine and cosine functions.

Since these angles are special angles from the unit circle, the values of each term can be determined from the x and y coordinate points at the specified angle.  

Solve each term and simplify the expression.

Find the value of .

Correct answer:

Explanation:

Using trigonometric relationships, one can set up the equation

.

Solving for ,

Thus, the answer is found to be 29.

Find the value of .

Correct answer:

Explanation:

Using trigonometric relationships, one can set up the equation

.

Plugging in the values given in the picture we get the equation,

.

Solving for ,

.

Thus, the answer is found to be 106.

Find all of the angles that satistfy the following equation:

Correct answer:

OR 

Explanation:

The values of  that fit this equation would be:

 and  

because these angles are in QI and QII where sin is positive and where

.

This is why the answer 

is incorrect, because it includes inputs that provide negative values such as:

Thus the answer would be each  multiple of  and  , which would provide the following equations:

  OR    

Evaluate: 

Correct answer:

Explanation:

To evaluate , break up each term into 3 parts and evaluate each term individually.

Simplify by combining the three terms.

What is the value of  ?

Correct answer:

Explanation:

Convert  in terms of sine and cosine.

Since theta is  radians, the value of  is the y-value of the point on the unit circle at  radians, and the value of  corresponds to the x-value at that angle.

The point on the unit circle at  radians is .  

Therefore,  and .  Substitute these values and solve.

Solve:  

Correct answer:

Explanation:

First, solve the value of .  

On the unit circle, the coordinate at  radians is .  The sine value is the y-value, which is .  Substitute this value back into the original problem.

Rationalize the denominator.

Find the exact answer for:  

Correct answer:

Explanation:

To evaluate , solve each term individually.

 refers to the x-value of the coordinate at 60 degrees from the origin.  The x-value of this special angle is .

 refers to the y-value of the coordinate at 30 degrees.  The y-value of this special angle is .

 refers to the x-value of the coordinate at 30 degrees.  The x-value is .

Combine the terms to solve .

Find the value of 

.

Correct answer:

Explanation:

The value of  refers to the y-value of the coordinate that is located in the fourth quadrant.

This angle  is also  from the origin.  

Therefore, we are evaluating .

Simplify the following expression:

Correct answer:

Explanation:

Simplify the following expression:

Begin by locating the angle on the unit circle. -270 should lie on the same location as 90. We get there by starting at 0 and rotating clockwise 

So, we know that 

And since we know that sin refers to y-values, we know that 

So therefore, our answer must be 1

All Precalculus Resources

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