How to Use Trigonometric Functions for Negative Angles (-θ): Identities & Examples

1 What Are Trigonometric Identities for Negative Angles?

Trigonometric functions are particularly important area of mathematics with numerous applications across various fields. Understanding how these functions change with respect to angles can be challenging, particularly for beginners. In this article, we will explain the following three identities related to negative angles:

\sin{(-\theta)} = -\sin{\theta} \tag{1}

\cos{(-\theta)} = \cos{\theta} \tag{2}

\tan{(-\theta)} = -\tan{\theta} \tag{3}

Assuming \theta > 0, these identities describe the properties of trigonometric functions when the angle is negative.

2 Key Identities for Negative Angles: Sine, Cosine, and Tangent

1. \sin{(-\theta)} = -\sin{\theta}:
   • For sine, the value flips sign when the angle changes from \theta to -\theta. For example, if \sin \theta = \frac{1}{2}, then \sin (-\theta) = -\frac{1}{2}.
2. \cos{(-\theta)} = \cos{\theta}:
   • Cosine remains unchanged when the angle is negated. The value of \cos (-\theta) is the same as that of \cos \theta.
3. \tan{(-\theta)} = -\tan{\theta}:
   • Tangent behaves like sine; its value reverses sign when the angle changes from \theta to -\theta.

3 Examples: Applying Negative Angle Trigonometric Identities

Let us practise applying these identities with specific examples.

Step 1: Remove the negative sign inside the sine

Using Equation 1,

\begin{aligned} \sin (-\theta) &= - \sin \theta\\ &= -\sin 30^\circ. \end{aligned}

Step 2: Calculate \sin 30^\circ

Here, \sin 30^\circ = \frac{1}{2}. If you are curious about why this is true, please refer to the definition of trigonometric functions.

Step 3: Substitute the value of \sin 30^\circ

Therefore,

\sin (-\theta) = -\sin 30^\circ = - \frac{1}{2},

so the answer is -\frac{1}{2}.

Step 1: Remove the negative sign inside the cosine

We want to find \cos (-\theta). Using Equation 2,

\begin{aligned} \cos (-\theta) &= \cos \theta\\ &= \cos 45^\circ. \end{aligned}

Step 2: Calculate \cos 45^\circ

\cos 45^\circ = \frac{1}{\sqrt{2}}. This can also be confirmed by the definition of trigonometric functions.

Step 3: Substitute \cos 45^\circ with its value

Thus,

\cos (-\theta) = \cos 45^\circ = \frac{1}{\sqrt 2}.

Step 4: Rationalise the denominator

When the denominator contains an irrational number such as \sqrt{2}, it is standard practice to rationalise the denominator. Since \sqrt{2} multiplied by itself equals 2, multiplying both the numerator and the denominator by \sqrt{2} removes the root from the denominator:

\begin{aligned} \frac{1}{\sqrt 2} &= \frac{1 \times \sqrt 2}{\sqrt 2 \times \sqrt 2}\\ &= \frac{\sqrt 2}{2}. \end{aligned}

Therefore, the answer is \frac{\sqrt 2}{2}.

Step 1: Remove the negative sign inside the tangent

Following the same approach as in Exercise 1 and Exercise 2, we have:

\tan (-\theta) = - \tan \theta = - \tan 60^\circ.

Step 2: Calculate \tan 60^\circ

By the definition of trigonometric functions, \tan 60^\circ = \sqrt 3.

Step 3: Substitute \tan 60^\circ with its value

Hence,

\tan (-\theta) = - \tan 60^\circ = - \sqrt 3.

4 Common Mistakes and Tips When Using Negative Angle Identities

In the identities introduced, sine and tangent keep a negative sign after transformation, whereas cosine does not. It is important to recognise that not all trigonometric functions follow the same rules in this respect.

Additionally, beginners—especially those who find mathematics difficult—often memorise identities without understanding them. This is not ideal. Memorising various identities can become burdensome and hinder the development of problem-solving skills. Many trigonometric identities can be derived through straightforward procedures, so it is beneficial to review their proofs.

5 Further Study of Negative Angle Trigonometric Identities

• Draw diagrams: In this article, we frequently used the definition of trigonometric functions when calculating values. This definition forms the foundation of trigonometry. If you are unfamiliar with it, I encourage you to review it.
• Practice problems: To deepen your understanding of these identities, consistently solve problems.