Lesson Video: Related and Correlated Angle Identities Mathematics

Video Transcript

In this video, we will learn how to write trigonometric functions like sine, cosine, and tangent and their reciprocals in terms of cofunctions and use their properties to recognize the equivalents of two trigonometric functions. We recall that if the angle 𝜃 is in standard position on a set of axes with a unit circle centered at the origin, then we can use the coordinates of the intersection between the terminal side of the angle and the unit circle to define sin 𝜃 and cos 𝜃. By looking at this in more detail, we will be able to define the sine and cosine of any angle.

Since tan 𝜃 is equal to sin 𝜃 over cos 𝜃, we can use these values to evaluate the tangent of any angle or any reciprocal trigonometric function. This geometric interpretation allows us to discover identities of the trig functions.

Let’s begin by considering the periodicity of these functions. A full rotation of 360 degrees or two 𝜋 radians counterclockwise of the angle 𝜃 will not change the position of its initial or terminal side. This will be true for any number of full rotations. Therefore, sin of 𝜃 is equal to sin of 𝜃 plus 360 multiplied by 𝑛. And the cos of 𝜃 is equal to the cos of 𝜃 plus 360 multiplied by 𝑛. As this also holds for any clockwise rotation, these identities are valid for any integer 𝑛.

When dealing in radians, we would replace 360 with two 𝜋. Since the tangent function is periodic every 180 degrees or 𝜋 radians, the tan of 𝜃 is equal to tan of 𝜃 plus 180 multiplied by 𝑛. This is once again true for any integer 𝑛, and we can replace 180 with 𝜋 when dealing in radians.

We can also take the reciprocal of both sides of any of these identities to find similar identities for the reciprocal trigonometric functions. These are the cosecant, secant, and cotangent, respectively. We can determine other trigonometric identities by using geometric transformations on the unit circle. For example, we can reflect angle 𝜃 in the 𝑦-axis. Since this is a reflection in the vertical axis, the two triangles are congruent. We can use this to determine the coordinates of the intersection between the hypotenuse and the unit circle. These are negative cos 𝜃, sin 𝜃.

We also notice that the hypotenuse is the terminal side of the angle 180 degrees minus 𝜃 in standard position. This means that the sin of 180 degrees minus 𝜃 is equal to the sin of 𝜃 and the cos of 180 degrees minus 𝜃 is equal to negative cos 𝜃. Using the fact that tan of 𝜃 is equal to sin of 𝜃 over cos of 𝜃, the tan of 180 degrees minus 𝜃 is equal to negative tan of 𝜃.

This leads us to a specific example of related angles, in this case when two angles sum to give 180 degrees or 𝜋 radians. Since these angles are known as supplementary angles, the three identities are known as the supplementary angle trigonometric identities. Once again, by taking the reciprocal of both sides, we can find similar identities for the reciprocal trigonometric functions.

We can follow the same process to find trigonometric identities of other related angles. Firstly, we have the point in the third quadrant with coordinates negative cos 𝜃, negative sin 𝜃 as shown. This corresponds to the angle 180 degrees plus 𝜃 in standard position. The sin of 180 degrees plus 𝜃 is therefore equal to negative sin 𝜃. And the cos of 180 degrees plus 𝜃 is equal to the negative of cos 𝜃. As dividing a negative by a negative gives a positive answer, the tan of 180 degrees plus 𝜃 is equal to the tan of 𝜃.

We can repeat this for the related angles 𝜃 and 360 degrees minus 𝜃. In this case, we have the sin of 360 degrees minus 𝜃 is equal to negative sin 𝜃. The cos of 360 degrees minus 𝜃 is equal to the cos of 𝜃. And the tan of 360 degrees minus 𝜃 is equal to the negative tan of 𝜃.

Identifying whether the sine, cosine, and tangent of any angle is positive or negative in each quadrant can be summarized using a CAST diagram. In the first quadrant, when 𝜃 lies between zero and 90 degrees, all three of our functions — the sine, cosine, and tangent — are positive. In the second quadrant, the sin of 𝜃 is positive, whereas the cos and tan of 𝜃 are negative. In the third quadrant, between 180 and 270 degrees, the sine and cosine functions are negative, whereas the tangent is positive. And finally, in the fourth quadrant, the sin of 𝜃 is negative, the cos of 𝜃 is positive, and the tan of 𝜃 is negative.

We will now look at an example where we’ll use the related angle identities to simplify a trigonometric expression.

Simplify the cos of 𝜃 plus the cos of 180 degrees minus 𝜃.

There are a few ways of simplifying this expression. For example, we could recall the supplementary angle identity. That tells us that the cos of 180 degrees minus 𝜃 is equal to negative cos of 𝜃. Substituting this into our expression, we would have the cos of 𝜃 plus negative cos of 𝜃. As this is the same as subtracting cos of 𝜃 from cos of 𝜃, our answer would be zero.

Whilst this method appears straightforward, it relies on us remembering the supplementary angle identity shown. There are many such identities, and trying to remember all these is difficult. It is therefore helpful to understand where these identities come from using the unit circle. If we sketch an acute angle 𝜃 in standard position, then the angle 180 degrees minus 𝜃 is the supplementary angle to 𝜃. Together, these form a straight line on the 𝑥-axis as shown.

Reflecting our triangle in the vertical axis, we have the angle 180 degrees minus 𝜃 in standard position. Since the triangles are congruent, their bases will be equal in length. This leads us to the identity the cos of 180 degrees minus 𝜃 is equal to the negative of cos of 𝜃. As this is true for any angle 𝜃 measured in degrees, we know that our answer is correct. cos of 𝜃 plus cos of 180 degrees minus 𝜃 is equal to zero.

Before looking at another example, we will consider complementary angles. We recall that complementary angles are angles which sum to 90 degrees. These are useful in trigonometry because if 𝜃 is one angle of a right triangle, then the other angle is 90 degrees minus 𝜃. We can use this information to find a set of identities called the cofunction identities. By firstly drawing 𝜃 in standard position, we can construct the following congruent triangle. This is the angle 90 degrees minus 𝜃 in standard position. Since the triangles are congruent, we can equate the corresponding sides such that the sin of 90 degrees minus 𝜃 equals cos 𝜃 and the cos of 90 degrees minus 𝜃 is sin 𝜃.

Using the fact that the tan of 𝜃 is equal to sin 𝜃 over cos 𝜃 and the cot of 𝜃 is the reciprocal of this, we have the tan of 90 degrees minus 𝜃 is equal to the cot of 𝜃. Once again, we can take the reciprocal of both sides of any of these identities to find identities for the reciprocal trigonometric functions.

The cofunction identities are one example of what are known as the corelated angle identities. The other corelated identities we will consider in this video are 𝜃 and 90 degrees plus 𝜃.

Returning to our unit circle, we will begin by rotating the triangle 90 degrees counterclockwise about the origin. This gives us a congruent triangle where the angle 90 degrees plus 𝜃 is in standard position. The sin of 90 degrees plus 𝜃 is therefore equal to cos 𝜃. The cos of 90 degrees plus 𝜃 is equal to negative sin of 𝜃. And the tan of 90 degrees plus 𝜃 is equal to negative cot of 𝜃. We could also develop this further for rotations of 180 and 270 degrees counterclockwise.

We will now look at one final example where we need to apply a cofunction identity.

Using the fact that the cos of 𝜃 is equal to the sin of 90 degrees minus 𝜃, which of the following is equivalent to the cos of 35 degrees? Is it (A) one over the sin of 35 degrees? (B) sin of 35 degrees. (C) sin of 55 degrees. (D) sin of 145 degrees. Or (E) negative sin of 35 degrees.

We can begin this question by substituting 𝜃 equals 35 degrees into both sides of our equation. This gives us the cos of 35 degrees is equal to the sin of 90 degrees minus 35 degrees. The right-hand side simplifies to the sin of 55 degrees, which from the five options given is option (C). This suggests that this is the correct answer. However, we can check by sketching the angle 35 degrees in standard position on the unit circle.

The coordinates of the point marked on the unit circle are the cos of 35 degrees, sin of 35 degrees. As this is a right triangle, the third angle is equal to 55 degrees. If we drew this angle of 55 degrees in standard position, it would be congruent to the triangle drawn for 35 degrees in standard position. The point on the unit circle has coordinates cos 55 degrees, sin 55 degrees. Since the height of this second triangle has the same length as the base of the first triangle, the sin of 55 degrees must be equal to the cos of 35 degrees. We can therefore conclude that option (C) is correct. If cos of 𝜃 is equal to the sin of 90 degrees minus 𝜃, then the cos of 35 degrees is equal to the sin of 55 degrees.

We will now summarize the key points from this video. We saw in this video that we can show properties of the trigonometric functions by considering the congruency of triangles formed by angles in standard position on the unit circle. In particular, we saw that the sin, cos, and tan of 180 degrees plus 𝜃 are equal to negative sin 𝜃, negative cos 𝜃, and tan 𝜃, respectively. We also had the corelated angle identities. These were the sin, cos, and tan of 90 degrees minus 𝜃 and 90 degrees plus 𝜃. All of these identities and more can be directly shown from the symmetries of the unit circle.