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,

Trigonometrical ratios

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



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