Vidyanti- Home

8.2-Trigonometric Ratios of Some Specific Angles

You are currently studying
Grade 10 → Math → Introduction to Trigonometry → 8.2-Trigonometric Ratios of Some Specific Angles

After successful completion of this topic, you should be able to:

Compute the trigonometric ratio of 0$^\circ$, 30$^\circ$, 45$^\circ$, 60$^\circ$, 90$^\circ$ in order to know and apply the value of specific angles.

In trigonometry, certain angles have specific values for their trigonometric ratios. These angles are commonly used and include $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$. The values of the trigonometric ratios for these angles can be derived from geometric principles, specifically using an equilateral triangle and an isosceles right triangle.

1. Trigonometric Ratios for $0^\circ$:

$\sin 0^\circ = 0$
$\cos 0^\circ = 1$
$\tan 0^\circ = 0$
$\csc 0^\circ = \text{undefined}$
$\sec 0^\circ = 1$
$\cot 0^\circ = \text{undefined}$

2. Trigonometric Ratios for $30^\circ$:

$\sin 30^\circ = \frac{1}{2}$
$\cos 30^\circ = \frac{\sqrt{3}}{2}$
$\tan 30^\circ = \frac{1}{\sqrt{3}}$
$\csc 30^\circ = 2$
$\sec 30^\circ = \frac{2}{\sqrt{3}}$
$\cot 30^\circ = \sqrt{3}$

3. Trigonometric Ratios for $45^\circ$:

$\sin 45^\circ = \frac{\sqrt{2}}{2}$
$\cos 45^\circ = \frac{\sqrt{2}}{2}$
$\tan 45^\circ = 1$
$\csc 45^\circ = \sqrt{2}$
$\sec 45^\circ = \sqrt{2}$
$\cot 45^\circ = 1$

4. Trigonometric Ratios for $60^\circ$:

$\sin 60^\circ = \frac{\sqrt{3}}{2}$
$\cos 60^\circ = \frac{1}{2}$
$\tan 60^\circ = \sqrt{3}$
$\csc 60^\circ = \frac{2}{\sqrt{3}}$
$\sec 60^\circ = 2$
$\cot 60^\circ = \frac{1}{\sqrt{3}}$

5. Trigonometric Ratios for $90^\circ$:

$\sin 90^\circ = 1$
$\cos 90^\circ = 0$
$\tan 90^\circ = \text{undefined}$
$\csc 90^\circ = 1$
$\sec 90^\circ = \text{undefined}$
$\cot 90^\circ = 0$

These specific values can be memorized to help solve various problems in trigonometry. Understanding these ratios is crucial for solving triangles and various applications in geometry, physics, and engineering.

One helpful tool for remembering these values is to use the acronym "All Students Take Calculus," which indicates the signs of the trigonometric functions in different quadrants:

All: All ratios are positive in the first quadrant.
Students: Sine is positive in the second quadrant.
Take: Tangent is positive in the third quadrant.
Calculus: Cosine is positive in the fourth quadrant.

By mastering these specific angle ratios and their properties, students can effectively apply trigonometric concepts in various mathematical contexts.

Solved Example: 8-2-03

If $\tan (A+B)= \sqrt{3}$ and $\tan (A–B)= \dfrac{1}{\sqrt{3}}$ ; 0°< A+B $\leq$ 90°; A > B, Find A and B.

Check Answer

Suggest improvements/Report errors

Solved Example: 8-2-04

State whether the following are true or false. Justify your answer.

(i) sin (A+B) = sin A + sin B.
(ii) The value of sin $\theta$ increases as $\theta$ increases.
(iii) The value of cos $\theta$ increases as $\theta$ increases.
(iv) sin $\theta$ = cos $\theta$ for all values of $\theta$.
(v) cot A is not defined for A=0°.

Check Answer

Solution:

(i) sin (A+B) = sin A + sin B.

False. The correct formula for $\sin(A + B)$ is $\sin(A + B) = \sin A \cos B + \cos A \sin B$.

(ii) The value of sin $\theta$ increases as $\theta$ increases.

False. The value of $\sin \theta$ increases from $0^\circ$ to $90^\circ$, but decreases from $90^\circ$ to $180^\circ$. It oscillates between -1 and 1.

(iii) The value of cos $\theta$ increases as $\theta$ increases.

False. The value of $\cos \theta$ decreases from $0^\circ$ to $90^\circ$ and then becomes negative for angles between $90^\circ$ and $270^\circ$.

(iv) sin$\theta$ = cos$\theta$ for all values of $\theta$.

False. $\sin \theta$ is equal to $\cos \theta$ only at specific angles, such as $\theta = 45^\circ$ or $\theta = 225^\circ$.

(v) cot A is not defined for A=0°.

True. The cotangent function $\cot A = \frac{1}{\tan A}$, and $\tan 0^\circ = 0$, so $\cot 0^\circ$ is undefined.