Evaluating Trigonometric Functions

Evaluating trigonometric functions means determining the values of sine, cosine, tangent, and their reciprocals for a given angle.

There are multiple ways to evaluate trigonometric functions: using right triangles, the unit circle, trigonometric identities, or a calculator.

For Special Angles

Certain angles in trigonometry appear frequently and are known to have well-known values. They are 0°, 30°, 45°, 60°, and 90°. They can also be written in radiants as ${0}$, ${\dfrac{\pi }{6}}$, ${\dfrac{\pi }{4}}$, ${\dfrac{\pi }{3}}$, and ${\dfrac{\pi }{2}}$

The values of sine, cosine, and tangent for these angles can be derived either using the unit circle or special right triangles (30°-60°-90° and 45°-45°-90° triangles).

Using Unit Circle

Each point (x, y) on the unit circle corresponds to an angle θ and has coordinates of (cos θ, sin θ). Below is the unit circle, along with the coordinates of the intersections. 

For example, from the unit circle, sin 45° = ${\sin \dfrac{\pi }{4}=\dfrac{\sqrt{2}}{2}}$ and cos 45° = ${\cos \dfrac{\pi }{4}=\dfrac{\sqrt{2}}{2}}$

Using Special Right Triangles

Special right triangles, such as the 45°-45°-90° triangle and the 30°-60°-90° triangle, provide exact values for sine, cosine, and tangent functions. The below table shows all the known values. 

For example, from the below table, sin 90° = 1

For Other Angles

Beyond the common angles, trigonometric functions can also be evaluated for any angle using reference angles, quadrant-specific sign rules, or identities.

Using Reference Angles

A reference angle is the acute angle (< 90°) that a given angle makes with the x-axis. It simplifies the evaluation of trigonometric functions for angles outside the first quadrant (> 90°). 

The sign of a function depends on the quadrant in which the angle lies (Quadrant-specific rules):

• Quadrant I (0° to 90°): All functions are positive.
• Quadrant II (90° to 180°): Only sine (sin) and cosecant (cosec⁡) are positive.
• Quadrant III (180° to 270°): Only tangent (tan) and cotangent (cot⁡) are positive.
• Quadrant IV (270° to 360°): Only cosine (cos⁡) and secant (sec) are positive.

By determining the reference angle and applying these quadrant rules, we can evaluate trigonometric functions easily.

Now, let us evaluate sin 150° 

The given angle, 150°, lies in the second quadrant. The reference angle is calculated as: 

180° – 150° = 30°

From the table of special angles:

sin 30° = ${\dfrac{1}{2}}$

Since in the second quadrant, sine (sin) is positive.

Thus, sin 150° = sin 30° = ${\dfrac{1}{2}}$

Using Trigonometric Identities

Here are the lists of the different types of trigonometric identities used to solve trigonometric functions. 

[Upload the diagram with the filename: Trigonometric Identities]

These identities simplify complex expressions and make evaluating functions easier.

Now, let us evaluate cos 75°

Using the angle sum identities,

cos 75° = cos(45° + 30°)

Using the cosine angle sum identities,

cos(45° + 30°) = cos 45° cos 30° – sin 45° sin 30°

Since cos 45° = sin 45° = ${\dfrac{1}{\sqrt{2}}}$, cos 30° = ${\dfrac{\sqrt{3}}{2}}$, and sin 30° = ${\dfrac{1}{2}}$

By substituting the known values into the formula,

cos 75° = ${\dfrac{1}{\sqrt{2}}\times \dfrac{\sqrt{3}}{2}-\dfrac{1}{\sqrt{2}}\times \dfrac{1}{2}}$

= ${\dfrac{\left( \sqrt{3}-1\right) }{2\sqrt{2}}}$

= ${\dfrac{\sqrt{2}\left( \sqrt{3}-1\right) }{4}}$

= ${\dfrac{\sqrt{6}-\sqrt{2}}{4}}$

Solved Examples