In this explainer, we will learn how to find the value of a trigonometric function from a given value of another trigonometric function.

Let us begin by recalling the trigonometric ratios and the unit circle. The unit circle is a circle with a radius of 1 whose center lies at the origin of a coordinate plane. For any point on the unit circle, a right triangle can be formed as shown in the following diagram. The hypotenuse of this right triangle makes an angle with the positive -axis.

Using right-angled trigonometry, we can define the trigonometric functions in terms of the unit circle:

We note that is not defined when .

The - and -coordinates of a point on the unit circle given by an angle are defined by and .

We also observe that while we have derived these definitions for an angle in quadrant 1, they hold for an angle in any quadrant. We know that to the right of the origin, the -values are positive, and to the left of the origin, the -values are negative. Similarly, above the origin, the -values are positive, and below the origin, the -values are negative.

If we add four points to our grid, , , , , where and are positive values, we see that they lie in each of the four quadrants.

At this stage, we also recall that the trigonometric functions cosecant , secant , and cotangent are the reciprocals of sine , cosine , and tangent such that

From the definition of the unit circle, we have

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 now consider a series of examples where we need to find the value of a trigonometric function from a given value of another trigonometric function.

In our first example, we need to calculate the sine of an angle, given both its cosine and tangent.

### Example 1: Finding the Value of the Sine of an Angle given the Values of the Tangent and Cosine

Find given and .

### Answer

We begin by recalling that since the tangent and cosine of our angle are both positive, then the angle must lie in the first quadrant such that . From our CAST diagram, we observe that the sine of the angle must also be positive.

As we can sketch a right triangle in the first quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 3, 4, and 5 such that .

Since then, from the diagram,

We can therefore conclude that if and , then .

For the remainder of the examples, we will need to use the reciprocal trigonometric identities.

### Example 2: Finding the Value of the Cosecant of an Angle given the Cotangent

If and , find .

### Answer

We begin by recalling that the cotangent of an angle is the reciprocal of the tangent of that angle such that

Since then

Using our knowledge of the CAST diagram, as and , we know that our angle lies in the fourth quadrant.

Since in the fourth quadrant and as the cosecant of an angle is the reciprocal of the sine of that angle, then .

As we can sketch a right triangle in the fourth quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 3, 4, and 5 such that .

Since then, from the diagram,

From the diagram, we see that . Using the properties of related angles, we know that

So,

As then,

We can therefore conclude that if and , then .

### Example 3: Finding the Cotangent of an Angle in a Specified Range given the Value of Sine

Find given , where .

### Answer

Using our knowledge of the CAST diagram, as , we know that our angle lies in the second quadrant.

Since in the second quadrant and as the cotangent of an angle is the reciprocal of the tangent of that angle, then .

As we can sketch a right triangle in the second quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 3, 4, and 5 such that .

Since then, from the diagram,

From the diagram, we see that and, using the properties of related angles, we know that

So,

As then,

We can therefore conclude that if , where , then .

### Example 4: Finding the Value of an Expression Using Trigonometric Equivalences

Find the value of , given , where is the smallest positive angle, and , where .

### Answer

Using our knowledge of the CAST diagram, as , where is the smallest positive angle, we know that lies in the first quadrant. Since , where , then lies in the third quadrant.

As we can sketch a right triangle in the first quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 3, 4, and 5 such that .

Since then, from the diagram,

As we can sketch a right triangle in the third quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 8, 15, and 17 such that .

Since then, from the diagram,

From the diagram, we see that and, using the properties of related angles, we know that

So,

Also,

So,

Substituting these values into our expression, we have

We can therefore conclude that if , where is the smallest positive angle, and , where , then .

### Example 5: Finding the Value of an Expression Using Trigonometric Equivalences

Find the value of , given that and .

### Answer

Using our knowledge of the CAST diagram, as , we know that lies in the third quadrant.

As we can sketch a right triangle in the third quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 3, 4, and 5 such that .

Since then, from the diagram,

From the diagram, we see that and, using the properties of related angles, we know that

So,

Also,

So,

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

So,

Substituting these values into our expression, we have

We can therefore conclude that if and , then .

### Example 6: Finding the Value of an Expression Involving Reciprocal Trigonometric Functions Using Trigonometric Equivalences

Find the value of given and .

### Answer

Using our knowledge of the CAST diagram, as , where and , we know that lies in the fourth quadrant.

As we can sketch a right triangle in the fourth quadrant. The right triangle is a Pythagorean triple consisting of three positive integers 3, 4, and 5 such that .

Since then, from the diagram,

From the diagram, we see that and, using the properties of related angles, we know that

So,

Also,

So,

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

So,

Substituting these values into our expression, we have

We can therefore conclude that if and , then .

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

### Key Points

- We can find the value of a trigonometric function from a given value of another trigonometric function by recalling the 6 trigonometric functions and the CAST diagram.
- Using the symmetries in the unit circle, we can also use the related-angle properties:
- These properties can be seen on the following diagram of the unit circle, where is the -coordinate and is the -coordinate.