In this explainer, we will learn how to evaluate the trigonometric functions with special angles and how to use them to evaluate trigonometric expressions.

We will begin by recalling special angles, together with the sine, cosine, and tangent values of these angles.

Let us consider a unit circle. This enables us to calculate , , and between and or 0 and radians. All three functions have key values at , , , , and . We know that , so , , and . Knowing these conversions enables us to solve trigonometric problems when the angles are given in either degrees or radians.

0 | 1 | 0 | 0 | ||

1 | 0 | 0 | 1 | ||

0 | undefined | 0 | undefined | 0 |

As the functions are periodic, we can calculate the sine, cosine, and tangent of angles outside of this range; however, this is not in the scope of this explainer.

Next, we recall that the special angles are , , and . The sine, cosine, and tangent of these angles are given below.

1 |

While we will not consider the derivation or proof of these results in this explainer, it is worth recalling the identity . This enables us to calculate the tangent of any angle if we are given the sine and cosine of that angle.

For example, since and , then .

### Definition: Reciprocal Trigonometric Functions

The reciprocal trigonometric functions cosecant , secant , and cotangent are the reciprocal of sine , cosine , and tangent such that

We can use these identities to calculate the cosecant, secant, and cotangent of , , and .

2 | |||

2 | |||

1 |

We will now recall the related angles of the trigonometric functions:

One way of recalling whether the sine, cosine, and tangent of any angle between and are positive or negative is using the CAST diagram. This is a memory device that we use to remember the signs of the trigonometric ratios in each of the four quadrants.

The Quadrant in Which the Terminal Side of the Angle Lies | The Interval in Which the Measure of the Angle Belongs | Signs of Trigonometric Functions | ||
---|---|---|---|---|

, | , | , | ||

First | + | + | + | |

Second | + | |||

Third | + | |||

Fourth | + |

We note that the angles are measured, from to or from 0 to radians in a counterclockwise direction, where the positive -axis is the initial side of the angle. The terminal side is where the angle stops. Any angle between and lies in the first quadrant. Any angle between and lies in the second quadrant. Any angle between and lies in the third quadrant. Any angle between and lies in the fourth quadrant.

### Example 1: Using Periodic Identities to Find the Value of a Trigonometric Function Involving Special Angles

Find the value of .

### Answer

We begin by recalling that .

So,

We therefore need to calculate .

Let us recall the property of related angles

If , then

From our knowledge of special angles, we know that .

So,

We can therefore conclude that .

### Example 2: Evaluating Trigonometric Expressions Involving Special Angles

Evaluate .

### Answer

We begin by recalling that .

So,

Also,

We therefore need to calculate .

From our knowledge of special angles, we know that and .

By considering the CAST diagram, as shown below, we see that lies in the 3rd quadrant and the sine of any angle here is negative.

Since then

So,

Substituting the values of and into our expression, we have

Therefore, .

In the remaining examples in this explainer, we will also need to use reciprocal trigonometric functions.

### Example 3: Using Periodic Identities to Evaluate a Trigonometric Expression Involving Special Angles

Find the value of .

### Answer

In order to answer this question, we need to recall reciprocal trigonometric identities and special angles.

From our knowledge of special angles, , , and .

Using the CAST diagram, we know that both the cosine and tangent of any angle in the second quadrant are negative. The sine and cosine of any angle in the third quadrant are also both negative.

The properties of related angles state that

So,

The properties of related angles also state that

So,

Therefore,

They also state that

So,

Therefore,

They also state that

So,

Therefore,

Since then

Substituting these values into our expression, we have

So, our final answer is .

### Example 4: Evaluating Trigonometric Expressions Involving Special Angles

Evaluate .

### Answer

Let us recall the property of related angles

If , then

From our knowledge of special angles, we know that .

So,

Substituting this value into our expression gives us

So, our final answer is 0.

### Example 5: Using Periodic Identities to Evaluate a Trigonometric Expression Involving Special Angles

Which of the following values of does **not** satisfy the equation ?

### Answer

In order to answer this question, we will use our knowledge of special angles and properties of related angles of the sine and cosine functions.

Let us begin by considering the five angles we are given and their corresponding position on the CAST diagram.

As , , and have the same position on the diagram, we know that the sine and cosine of these answers will be equal.

For example, we know that .

So,

Likewise, .

So,

We therefore need only consider the three options A, B, and C.

Firstly, let us consider . Substituting this into the left-hand side of our equation, we have

We recall the property of related angles

If , then

Substituting and into this expression gives us

Secondly, let us consider . Substituting this into the left-hand side of our equation, we have

We recall the property of related angles

If , then

Substituting and into this expression gives us

Finally, let us consider . Substituting this into the left-hand side of our equation, we have

We recall the property of related angles

If , then

Substituting and into this expression gives us

We can therefore conclude that does **not** satisfy the equation , whereas all other four options do.

### Example 6: Evaluating Trigonometric Expressions Involving Special Angles

Find the value of .

### Answer

In order to answer this question, we need to recall reciprocal trigonometric identities and special angles.

From our knowledge of special angles, we know that , , and .

From the sine and cosine graphs below, we see that and .

Since then

We can now substitute all of these values into our expression:

We will finish this explainer by recapping some of the key points.

### Key Points

- We can evaluate trigonometric functions and expressions using our knowledge of special angles:
- We can use reciprocal trigonometric identities to solve more complicated problems:
- We can also use properties of related angles to evaluate trigonometric expressions: