When the sum of two angles equals 90^\circ, these angles are called “complementary angles.” There are several useful identities in trigonometry related to complementary angles:
\sin{\left(\frac{\pi}{2} - \theta \right)} = \cos{\theta} \tag{1}
\cos{\left(\frac{\pi}{2} - \theta \right)} = \sin{\theta} \tag{2}
\tan{\left(\frac{\pi}{2} - \theta \right)} = \frac{1}{\tan{\theta}} \qquad (\tan \theta \ne 0) \tag{3}
Since the sum of \frac{\pi}{2} - \theta and \theta is \frac{\pi}{2} (90^\circ), these two angles are complementary. The identities above hold true for complementary angles and are therefore called “complementary angle identities.”
In this article, we will explain the characteristics of these identities and how to apply them.
1 Characteristics of Each Identity
1.1 \sin{\left(\frac{\pi}{2} - \theta \right)} = \cos{\theta}
This identity expresses that “the sine of a complementary angle equals the cosine of the original angle.” Specifically, considering the sine of \frac{\pi}{2} - \theta on the left-hand side corresponds exactly to the cosine of \theta on the right-hand side.
A visual representation of this claim is shown in Figure 1.
Although trigonometric functions are formally defined using the unit circle (more details here), we will here consider a right-angled triangle for simplicity. In the orange triangle ABC shown in Figure 1, the lengths of the sides are AB = r, BC = x, and CA = y. Given that \angle ABC = \theta, the triangle’s interior angles sum to 180^\circ, so \angle CAB = 90^\circ - \theta = \frac{\pi}{2} - \theta.
Looking at sine in this orange triangle, we find:
\sin \theta = \frac{y}{r}
Here, we focused on angle \theta. Now, consider the purple triangle in Figure 1, which is congruent to the orange triangle and has \frac{\pi}{2} - \theta, with vertices corresponding as follows:
\begin{aligned} A &\rightarrow A'\\ B &\rightarrow B'\\ C &\rightarrow C' \end{aligned}
Focusing on the angle \angle C'A'B' in the purple triangle, we examine the cosine:
\cos \left(\frac{\pi}{2} - \theta\right) = \frac{y}{r}
This matches the previous expression for \sin \theta. Therefore, we confirm:
\sin \theta = \cos \left(\frac{\pi}{2} - \theta\right)
1.2 \cos{\left(\frac{\pi}{2} - \theta \right)} = \sin{\theta}
Just like Equation 1, this identity shows that cosine transforms into sine when dealing with complementary angles. Please use Figure 1 to explore and deepen your understanding of this relationship.
1.3 \tan{\left(\frac{\pi}{2} - \theta \right)} = \frac{1}{\tan{\theta}}
This identity states that the tangent of a complementary angle is the reciprocal of the tangent of the original angle \theta. You can also visualise this using Figure 1, so please take the time to consider it carefully.
2 Practice Using the Identities
Let us now solve some problems by applying these identities.
Exercise 1 Express the following trigonometric functions using angles between 0^\circ and 45^\circ.
- \sin 55^\circ
- \cos 78^\circ
- \tan 82^\circ
Solution 1. Applying the identities directly allows us to solve these.
Step 1: Select the appropriate identity
Since we want to rewrite sine, choose the identity, including sine, Equation 1:
\sin{\left(\frac{\pi}{2} - \theta \right)} = \cos{\theta}
Step 2: Rewrite \sin 55^\circ
Express \sin 55^\circ using 90^\circ as a sum (or a difference):
\sin 55^\circ = \sin (90^\circ - 35^\circ).
Note that 90^\circ \Leftrightarrow \frac{\pi}{2} and 90 - 35 = 55.
Step 3: Apply the identity
Using the identity, we find:
\begin{aligned} \sin 55^\circ &= \sin (90^\circ - 35^\circ)\\ &= \cos 35^\circ. \end{aligned}
Since 35^\circ lies between 0^\circ and 45^\circ, the answer is \cos 35^\circ.
Step 1: Select the appropriate identity
For cosine, we use the identity Equation 2:
\cos{\left(\frac{\pi}{2} - \theta \right)} = \sin{\theta}
Step 2: Rewrite \cos 78^\circ
Express \cos 78^\circ with 90^\circ:
\cos 78^\circ = \cos (90^\circ - 12^\circ).
Step 3: Apply the identity
Applying the identity:
\begin{aligned} \cos 78^\circ &= \cos (90^\circ - 12^\circ)\\ &= \sin 12^\circ. \end{aligned}
So the answer is \sin 12^\circ.
Step 1: Select the appropriate identity
For tangent, use the identity Equation 3:
\tan{\left(\frac{\pi}{2} - \theta \right)} = \frac{1}{\tan{\theta}}
Step 2: Rewrite \tan 82^\circ
Rewrite with 90^\circ:
\tan 82^\circ = \tan (90^\circ - 8^\circ).
Step 3: Apply the identity
Applying the identity:
\begin{aligned} \tan 82^\circ &= \tan (90^\circ - 8^\circ)\\ &= \frac{1}{\tan 8^\circ}. \end{aligned}
Therefore, the answer is \frac{1}{\tan 8^\circ}.
3 Cautions When Using These Identities
When using each identity, it is crucial to identify the complementary angle. The complementary angle when the original angle is \theta is 90^\circ - \theta = \frac{\pi}{2} - \theta.
Be sure to keep the units consistent when calculating values, using either degrees or radians. For example, \sin 72^\circ = \sin (90^\circ - 18^\circ), or \cos \frac{2\pi}{5} = \cos \left(\frac{\pi}{2} - \frac{\pi}{10} \right). Mixing units can cause errors.
Note that for the tangent identity Equation 3, the identity is invalid when \theta = 0^\circ. In this case,
\tan \left( \frac{\pi}{2} - \theta \right) = \tan \frac{\pi}{2}
does not exist, so the identity does not hold.
4 Tips for Future Learning
Understanding trigonometric functions well requires not only memorising identities and properties but also grasping their geometric meanings. Here are some tips:
Draw diagrams: Visualising trigonometric identities deepens understanding. For example, try drawing Figure 1 yourself to derive the identities.
Practice repeatedly: Solve problems using these identities to gain confidence. At first, feel free to refer to the identities, and gradually learn when and how to apply each one.
