Trigonometry Formulas with Examples and FAQs

Trigonometry Formulas

Trigonometry finds its use in establishing relations between angles, lengths of the sides of any triangle and height of any triangle. We generally associate it with right-angled triangles. One such triangle is shown below, to establish relations between the angle θ and the sides, i.e., perpendicular(the side opposite to θ), base(the side adjacent to θ) and hypotenuse(the side opposite to 90°) in case of a right-angled triangle.

Basic Trigonometric Formulas

The basic trigonometric ratios are namely sine(sin), cosine(cos), tangent(tan), cosecant(cosec), secant(sec) and cotangent(cot). The trigonometry ratios for the angle θ with respect to the above figure is:

1. \sin \theta=\frac{\text { perpendicular }}{\text { hypotemuse }}

2. \cos \theta=\frac{\text { base }}{\text { hypotenuse }}

3. \tan \theta=\frac{\text { perpendicular }}{\text { base }}

4. \operatorname{cosec} \theta=\frac{\text { hypotemuse }}{\text { perpendicular }}

5. \sec \theta=\frac{\text { hypotenuse }}{\text { base }}

6. \cot \theta=\frac{\text { base }}{\text { perpendicular }}

Reciprocal Relations between Trigonometric Ratios

1. \operatorname{cosec} \theta=\frac{1}{\sin \theta}

2. \sec \theta=\frac{1}{\cos \theta}

3. \cot \theta=\frac{1}{\tan \theta}

Relations between Trigonometric Ratios

1. \tan \theta=\frac{\sin \theta}{\cos \theta}

2. \cot \theta=\frac{\cos \theta}{\sin \theta}

Trigonometric Ratios of Complementary Angles

1. \sin \left(90^{\circ}-\theta\right)=\cos \theta

2. \cos \left(90^{\circ}-\theta\right)=\sin \theta

3. \tan \left(90^{\circ}-\theta\right)=\cot \theta

4. \operatorname{cosec}\left(90^{\circ}-\theta\right)=\sec \theta

5. \sec \left(90^{\circ}-\theta\right)=\operatorname{cosec} \theta

6. \cot \left(90^{\circ}-\theta\right)=\tan \theta

Table for Values of Trigonometric Ratios for standard angles (0° to 90°)

Trigonometric Identities

1. \sin ^{2} \theta+\cos ^{2} \theta=1

2. \tan ^{2} \theta+1=\sec ^{2} \theta

3. \cot ^{2} \theta+1=\operatorname{cosec}^{2} \theta

Examples

Example 1: If the value of \sin \theta=\frac{12}{13}, find all the remaining trigonometric ratios.

Solution:

Given, the value of \sin \theta=\frac{12}{13},

We know, \sin \theta=\frac{\text { perpendicular }}{\text { hypotenuse }}

Perpendicular = 12 units and Hypotenuse = 13 units.

By Pythagoras theorem for right triangle we have,

\text { Perpendicular }^{2}+\text { base }^{2}=\text { hypotenuse }^{2}

\Rightarrow \text { Base }=\sqrt{\text { Hypotenuse }^{2}-\text { Perpendicular }^{2}}

Substituting Value of perpendicular and hypotenuse we get:

\text { Base }=\sqrt{13^{2}-12^{2}}=\sqrt{169-144}=\sqrt{25}=5 \text { units }

The trigonometry ratios for the angle θ with respect to the given value of sin are:

1. \cos \theta=\frac{\text { base }}{\text { hypotenuse }}=\frac{5}{13}
2. \tan \theta=\frac{\text { perpendicular }}{\text { base }}=\frac{12}{5}
3. \operatorname{cosec} \theta=\frac{\text { hypotense }}{\text { perpendicular }}=\frac{13}{12}
4. \sec \theta=\frac{\text { hypotenuse }}{\text { base }}=\frac{13}{5}
5. \cot \theta=\frac{\text { base }}{\text { perpendicular }}=\frac{5}{12}

Example 2: Find the value of \frac{\sin 30^{\circ}}{\tan 45^{\circ}}+\cos ^{2}\left(60^{\circ}\right)

Solution:  

\sin 30^{\circ}=\frac{1}{2}, \tan 45^{\circ}=1 \text { and } \cos 60^{\circ}=\frac{1}{2}

\therefore  \frac{\sin 30^{\circ}}{\tan 45^{\circ}}+\cos ^{2}\left(60^{\circ}\right)
=\frac{\frac{1}{2}}{1}+\left(\frac{1}{2}\right)^{2}
=\frac{1}{2}+\frac{1}{4}
=\frac{2+1}{4}
=\frac{3}{4}

Hence, \frac{\sin 30^{\circ}}{\tan 45^{\circ}}+\cos ^{2}\left(60^{\circ}\right)=\frac{3}{4}