# Trigonometric Ratios

Trigonometric Ratios

Trigonometric is a branch of mathematics that deals with the study of triangle and it’s relationship between its sides.

Some of the important formulas used in trigonometric ratios are listed below:-

Let, $\theta$ be the reference angle, then side AB = P opposite to$\theta$ is called perpendicular, the side BC = b, adjacent to $\theta$ is called the base. The side AC = h, opposite to right angle is called the hypotenuse.

Hence,

p = Perpendicular

b = Base $\theta$

h = Hypotenuse

now, the six trigonometrical ratios, we have

1.            $\sin\theta = \dfrac{p}{h}$

2.            $\cos\theta = \dfrac{b}{h}$

3.            $\tan\theta = \dfrac{p}{b}$

4.            $\csc\theta = \dfrac{h}{p}$

5.            $\sec\theta = \dfrac{h}{b}$

6.            $\cot\theta = \dfrac{b}{p}$

Formulas

1.

a. $\sin\theta \times \csc\theta =1$

b. $\sin\theta = \dfrac{1}{\csc\theta}$

c. $\csc\theta = \dfrac{1}{\sin\theta}$

2.

a. $\cos\theta \times \sec\theta = 1$

b. $\cos\theta = \dfrac{1}{\sec\theta}$

c. $\sec\theta = \dfrac{1}{\cos\theta}$

3.

a. $\tan\theta \times \cot\theta = 1$

b. $\tan\theta = \dfrac{1}{\cot\theta}$

c. $\cot\theta = \dfrac{1}{\tan\theta}$

4.

a. $\sin^2 \theta + \cos^2 \theta = 1$

b. $\sin^2\theta = 1- \cos^2\theta$

c. $\cos^2 \theta = 1- \sin^2 \theta$

d. $\sin\theta =\sqrt{1- cos^2 \theta}$

e. $\cos\theta = \sqrt{1- \sin^2\theta}$

5.

a. $\sec^2\theta = 1 + \tan^2 \theta$

b. $\tan^2 \theta = \sec^2 \theta- 1$

c. $\sec^2\theta- \tan^2 \theta = 1$

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

e. $\tan\theta = \sqrt{\sec^2 \theta- 1}$

6.

a. $\csc^2 \theta =1 + \cot^2\theta$

b. $\cot^2 \theta = \csc^2 \theta- 1$

c. $\csc^2 \theta- \cot^2 \theta = 1$

d. $\csc\theta = \sqrt{1 + \cot^2 \theta}$

e. $\cot\theta = \sqrt{\csc^2 \theta- 1}$

7. $\tan\theta = \dfrac{sin\theta}{\cos\theta}$

8. $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

TRIGONOMETRIC RATIOS OF ALLIED ANGLES:

For $0^\circ , 180^\circ \& 360^\circ$, all ratios are unchanged.

For $90^\circ, 270^\circ$ all ratios are changed to $\cdots \cos \cdots \sin, \tan \cdots \cot, \csc \cdots \sec$

Trigonometrical ratios of angles of any magnitude and sign

1.            Ratios of $(90^\circ- \theta)$

Since $(90^\circ- \theta)$lies in First Quadrant So By the concept of (CAST) all positive

a.            $\sin(90^\circ- \theta) = \cos\theta$

b.            $\cos(90^\circ- \theta) = \sin\theta$

c.            $\tan(90^\circ- \theta) = \cot\theta$

d.            $\cot(90^\circ- \theta) = \tan\theta$

e.            $\sec(90^\circ- \theta) = \csc\theta$

f.             $\csc(90^\circ- \theta) = \sec\theta$

Note: The angles $\theta$ and $(90^\circ- \theta)$ are called complementary angles and each is called complement of other angles.

2.            Ratios of $(-\theta)$

Since $(-\theta)$ lies in fourth Quadrant so $\cos / \sec$ positive

a.            $\sin(90^\circ- \theta) = \cos\theta$

b.            $\cos(90^\circ- \theta) = \sin\theta$

c.            $\tan(90^\circ- \theta) = \cot\theta$

d.            $\cot(90^\circ- \theta) = \tan\theta$

e.            $\sec(90^\circ- \theta) = \csc\theta$

f.             $\csc(90^\circ- \theta) = \sec\theta$

Note: $\cos\theta \& \sec\theta$ can’t change when $\theta$ is replaced by $- \theta$, but other four trigonometric ratios change their sign, but not their magnitude.

3.            Ratios of $(90^\circ + \theta)$

Since $(90^\circ + \theta$ lies in second Quadrant so $\sin$ and $\csc$ positive.

a.            $\sin (90^\circ + \theta) = \cos\theta$

b.            $\cos(90^\circ + \theta) = -\sin\theta$

c.            $\tan(90^\circ + \theta) = -\cot\theta$

d.            $\cot(90^\circ + \theta) = \tan\theta$

e.            $\sec(90^\circ + \theta) = -\csc\theta$

f.             $\csc(90^\circ + \theta) = \sec\theta$

4.            Ratios of $(180^\circ- \theta)$

Since $(180^\circ- \theta)$ also lies in second quadrant so $\sin$ and $\csc$.

a.            $\sin(180^\circ- \theta) = \sin\theta$

b.            $\cos(180^\circ- \theta) = -\cos\theta$

c.            $\tan(180^\circ- \theta) = -\tan\theta$

d.            $\cot(180^\circ- \theta) = -\cot\theta$

e.            $\sec(180^\circ- \theta) = -\sec\theta$

f.             $\csc(180^\circ- \theta) = \csc\theta$

Note: The angle $\theta$ and $(180^\circ- \theta)$ are called supplementary angle and each is called supplement of other.

5.            Ratio of $(180 + \theta)$

a.            $\sin(180^\circ + \theta) = -\sin\theta$

b.            $\cos(180^\circ + \theta) = -\cos\theta$

c.            $\tan(180^\circ + \theta) = \tan\theta$

d.            $\cot(180^\circ + \theta) = \cot\theta$

e.            $\sec(180^\circ + \theta) = -\sec\theta$

f.             $\csc(180^\circ + \theta) =- \csc\theta$

6.            Ratios of $(270^\circ- \theta)$

Since $(270^\circ- \theta)$ lies in third quadrant so $\tan$ and $\cot$ positive

a.            $\sin(270^\circ- \theta) = -\cos\theta$

b.            $\cos(270^\circ- \theta) = \sin\theta$

c.            $\tan(270^\circ- \theta) = \cot\theta$

d.            $\cot(270^\circ- \theta) = \tan\theta$

e.            $\sec(270^\circ- \theta) = -\csc\theta$

f.             $\csc(270^\circ- \theta) =-\sec\theta$

7.            Ratios of $(270^\circ + \theta)$

Since $(270^\circ + \theta)$ lies in fourth quadrant so $\cos, \sec$ positive

a.            $\sin(270^\circ + \theta) = -\cos\theta$

b.            $\cos(270^\circ + \theta) = \sin\theta$

c.            $\tan(270^\circ + \theta) = -\cot\theta$

d.            $\cot(270^\circ + \theta) = -\tan\theta$

e.            $\sec(270^\circ + \theta) = \csc\theta$

f.             $\csc(270^\circ + \theta) =-\sec\theta$

8.            Ratios of $(360^\circ- \theta)$

Since $(360^\circ- \theta)$ lies in fourth quadrant so $\cos, \sec$ positive.

a.            $\sin(360^\circ- \theta) = -\sin\theta$

b.            $\cos(360^\circ- \theta) = \cos\theta$

c.            $\tan(360^\circ- \theta) = -\tan\theta$

d.            $\cot(360^\circ- \theta) = -\cot\theta$

e.            $\sec(360^\circ- \theta) = \sec\theta$

f.             $\csc(360^\circ- \theta) = -\csc\theta$

Note: Ratio of $(360^\circ- \theta)$ and those $(-\theta)$ are same

9.            Ratio of $(360^\circ + \theta)$

Since Ratio of $(360^\circ + \theta)$ lies in first quadrant so all positive

a.            $\sin(360^\circ + \theta) = \sin\theta$

b.            $\cos(360^\circ + \theta) = \cos\theta$

c.            $\tan(360^\circ + \theta) = \tan\theta$

d.            $\cot(360^\circ + \theta) = \cot\theta$

e.            $\sec(360^\circ + \theta) = \sec\theta$

f.             $\csc (360^\circ + \theta) =\csc\theta$

Related posts:

1. Derivatives of Trigonometric functions. As you know, The functions SINE x(sin x) , CO-SECANT...
2. Derivatives of inverse trigonometric functions Inverse trigonometric functions  are the  inverse of trigonometric functions ....
3. Trigonometric functions of negative angles Trigonometric functions of negative angles. How to find trigonometric functions...
4. Pythagorian Identities Fundamental Pythagorian identity of trigonometry and other basic trigonometric formulas...
5. Integration by trigonometric substitution integration by trigonometric substitution: One of the most powerful techniques...