Lesson Explainer: Evaluating Reciprocal Trigonometric Functions Mathematics

In this explainer, we will learn how to evaluate reciprocal trigonometric functions at given values in both degrees and radians.

Recall that 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 in the following diagram.

The hypotenuse of the right triangle forms an acute angle, which we label , with the positive . Its length is 1, the same as the radius of the unit circle, and it goes from the origin to point . This point then defines the lengths of the two shorter sides. The side opposite the angle has length . The adjacent side has length .

In questions on this topic, we will meet some angles that are measured in radians instead of degrees. We can convert between radians and degrees, as needed, using the fact that .

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

Since and , the tangent function can also be written in terms of sine and cosine:

On the unit circle, the standard angle is measured from to in a counterclockwise direction, starting from the positive , which we refer to as the initial side of the angle. We refer to the side where the standard angle stops as the terminal side.

We find satisfying in quadrant I of the unit circle; that is, in radians.

We find satisfying in quadrant II of the unit circle; that is, in radians.

We find satisfying in quadrant III of the unit circle; that is, in radians.

We find satisfying in quadrant IV of the unit circle; that is, in radians.

Quadrantal angles terminate along the - or , specifically where equals , , , , or . We should note that is coterminal with , since both of these angles terminate on the positive . In radians, these quadrantal angles are 0, , , , and .

What we have looked at so far appears to be a departure from foundational right triangle trigonometry, where is always an acute angle and triangle sides are measured in positive units. But when we refer to the unit circle, we can still make use of right triangle trigonometry concepts if we think of and as the adjacent and opposite sides of a reference triangle. A reference triangle is a right triangle created by dropping a perpendicular segment from point on the unit circle back to the . It is important to recognize that on the unit circle, and can take any real number value including zero or a negative.

Let’s examine the angle whose vertex is at and whose sides are the and the hypotenuse of a reference triangle. This is called a reference angle . Therefore, even if , will be an acute angle. The following diagram shows the placement of reference triangles for all four quadrants.

Given a standard angle on the unit circle, the corresponding reference angle can be calculated for any quadrant as follows:

In quadrant I,

In quadrant II,

In quadrant III,

In quadrant IV,

In the diagram below, we show the unit circle with every multiple of from to . The radian conversions are included in pink.

Now, recall the properties of -- special right triangles, as shown in the next diagram. According to these properties, a -- right triangle with a hypotenuse of 1 will have shorter sides of length and . The hypotenuse of 1 corresponds to the radius of a unit circle.

Dividing the unit circle into increments results in eight -- reference triangles.

Thus, each coordinate value is , , , or , depending on which quadrant we are in. The exact coordinates are written as ordered pairs for each multiple of .

We can also find every multiple of from to on the unit circle, in degrees or radians.

Now, recall the properties of -- special right triangles, as shown in the diagram below. According to these properties, a -- right triangle with a hypotenuse of 1 will have shorter sides, both of length .

Dividing the unit circle into increments gives us four -- reference triangles.

Thus, each coordinate is either or , depending on which quadrant we are in. The exact coordinates are included in the following diagram.

We can determine the sign of our trigonometric function value based on where the angle terminates. One way of recalling whether the sine, cosine, and tangent of any angle between and are positive or negative is by using the CAST diagram. This is a memory device that we use to remember the signs of the trigonometric functions in each of the four quadrants.

• In quadrant I, All values are positive.
• In quadrant II, Sine is positive.
• In quadrant III, Tangent is positive.
• In quadrant IV, Cosine is positive.

Now that we have reviewed properties of standard trigonometric functions in relation to the unit circle, we will turn our attention to the reciprocals of these functions. Note that the sign of the value does not change when we take the reciprocal.

We will use the earlier established sine, cosine, and tangent definitions, described in terms of the unit circle, to write new definitions of their reciprocals.

Given the trigonometric function , its reciprocal function, cosecant, can be written as

Given the trigonometric function , its reciprocal function, secant, can be written as

Given the trigonometric function , its reciprocal function, cotangent, can be written as

Now, we will practice using the CAST diagram to determine the sign of various trigonometric functions.

Example 1: Identifying the Sign of a Reciprocal Trigonometric Ratio

Which of the following trigonometric functions is positive at ?

1. Secant
2. Cosecant
3. Cotangent
4. None of the answers are correct.

Answer

To answer this question, we will need to find on the unit circle and identify the signs of standard trigonometric functions in that particular quadrant. If desired, we can convert radians into degrees, using the fact that :

In quadrant I, all trigonometric functions are positive, due to the positive - and -coordinates. But we find in quadrant III, where only tangent and its reciprocal, cotangent, are positive. This information comes from the CAST diagram, which is a memory device to help us recall which standard trigonometric functions are positive in each quadrant.

• In quadrant I, All values are positive.
• In quadrant II, Sine is positive.
• In quadrant III, Tangent is positive.
• In quadrant IV, Cosine is positive.

The reciprocal trigonometric functions are defined in terms of the standard trigonometric functions as follows:

The sign of the value does not change when we take the reciprocal, and we know that only tangent and its reciprocal are positive in quadrant III, where (equivalently, radians) is found.

Therefore, of the three reciprocal trigonometric functions, only cotangent is positive at .

This means that choice C is correct.

Next, we will look at how to find the exact value of a reciprocal trigonometric expression when it involves an angle in radians that can be found on the unit circle.

Example 2: Finding the Exact Value of a Reciprocal Trigonometric Expression

Find the exact value of .

Answer

To begin, we find the standard angle in quadrant II of the unit circle. If desired, we can convert radians into degrees, using the fact that :

We recall that cotangent is the reciprocal of the trigonometric function tangent. Therefore, . For any point on the unit circle, and . So, we need to find both the - and -coordinates of the point where the standard angle intersects the unit circle.

Next, we calculate the reference angle that makes with the negative . This angle is found by subtracting from . We can then sketch a special -- right reference triangle, as shown in the diagram below.

Now, we recall the properties of -- special right triangles, as shown in the next diagram. According to these properties, a -- right triangle with a hypotenuse of 1 will have shorter sides of length and . The hypotenuse of 1 corresponds to the radius of a unit circle.

On the unit circle, the -coordinate of the point we desire corresponds to the length of the side adjacent to the reference angle. The -coordinate corresponds to the length of the side opposite the reference angle. However, because of the location of the reference triangle in quadrant II, the -coordinate must be negative. Thus, the terminal side of the angle intersects the point .

Since , we can substitute the - and -coordinates to find the exact value.

This means we will need to simplify the resulting complex fraction:

Recall how taking the quotient of two fractions is more simply calculated by multiplying the first fraction by the reciprocal of the second fraction. In this case, we multiply by the reciprocal of and simplify. So,

In conclusion, the exact value of or is .

As we reviewed earlier in this explainer, the unit circle only provides the - and -coordinates for angles with measurements equal to multiples of or between and . For other angle measures not found on the unit circle, we can use a calculator to find the approximate value of trigonometric functions. In cases like this, we will usually be asked to round to a specific number of significant figures. We will need a calculator that can switch between degree mode and radian mode.

For example, we can place in standard position on a coordinate plane in the second quadrant. But it would not match any of the designated angles on the unit circle because it is not a multiple of or . The following diagrams show how lies between two angles given on the unit circle: and .

Now, let’s see how to find the approximate secant value at using our calculator.

Next, let’s consider how we would evaluate trigonometric functions at quadrantal angles, which terminate along the - or , as shown in the next diagram.

We need to pay special attention to coordinate points containing zero. This presents a unique challenge because we cannot possibly draw a reference triangle with a side length of zero. This means some trigonometric functions are not defined at specific quadrantal angles. This occurs whenever a coordinate of 0 appears in the denominator of the function. The angles where the trigonometric functions are undefined are highlighted in the table below.

In the following example, we will look at how to find the exact value of a reciprocal trigonometric function of a negative quadrantal angle. Although negative angles are not shown on the unit circle, we will use our knowledge of coterminal angles to find an equivalent principal angle. Recall that coterminal angles share the same initial and terminal sides. Performing any number of full rotations, clockwise or counterclockwise, will lead us to a coterminal angle.

Example 4: Finding the Exact Value of a Reciprocal Trigonometric Expressio

Find the exact value of .

Answer

Let’s begin by imagining that the standard angle is the angle between two rays and , where is the initial side and is the terminal side of the angle.

As our angle is negative, we need to measure the angle in a clockwise direction from the initial side , as shown.

We recall that coterminal angles share the same initial and terminal sides. There is an infinite number of equivalent coterminal angles. However, we need to find a principal angle equivalent to , which should terminate between and . To find this angle, we need to measure in the counterclockwise positive direction.

We recall that angles at a point sum to (or if in radians), so a positive equivalent of is found by subtracting from :

Therefore, the principal angle equivalent to is . As seen on the unit circle, is a quadrantal angle intersecting the point .

We are looking for , which we now know is equivalent to . We recall that cosecant is the reciprocal of the trigonometric function sine. For any point on the unit circle, ; therefore, . Using the point , we have .

Since we have seen that is coterminal to , then .

In conclusion, we have shown that .

Now that we have practiced evaluating various reciprocal trigonometric functions in degrees and radians, let’s try evaluating a product.

Example 5: Evaluating a Reciprocal Trigonometric Expression

Evaluate .

Answer

To evaluate , we need to find the cosecant and cotangent values first and then work out their product. We recall that cosecant is the reciprocal of sine, and cotangent is the reciprocal of tangent.

We find that lies in the third quadrant of the unit circle, as shown above. The terminal side of the angle intersects the point . Since sine is defined as the -coordinate, we have

Next, we will take the reciprocal of to find . The reciprocal of is found by swapping the numerator and denominator, so

We do not need to be concerned about rationalizing the denominator at this point, since this is not our final answer. Next, we need to find .

Cotangent is the reciprocal of tangent. Since , we will be using . We refer to the point we found earlier and substitute the - and -coordinates into the cotangent formula:

Recall how taking the quotient of two fractions is more simply calculated by multiplying the first fraction by the reciprocal of the second fraction. In this case, we multiply by the reciprocal of and simplify. So,

Finally, we use substitution to evaluate the original expression and work out the product:

Let’s finish by recapping some important points from the explainer.

Key Points

• The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The standard angles found on the unit circle are multiples of and between and . The coordinate pairs corresponding to these angles are derived from the sides of special right triangles with a hypotenuse length of 1.

• The reciprocal trigonometric functions, cosecant, secant, and cotangent, are defined in terms of the standard trigonometric functions as follows:
• All trigonometric functions can also be defined using points from the unit circle.

	, where
	, where
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