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# Trigonometry

## Trigonometry Synopsis

Synopsis

Important Concepts

• An angle which has a magnitude as well as the direction of rotation is known as directed angle.
• An angle is formed by two rays called initial ray and terminal ray, with a common initial point called vertex.
• If the rotation from the initial ray to the terminal ray is anti-clockwise it is to be taken as positive and if it is clockwise, rotation is to be taken as negative.
• Each trigonometrical ratio is a real number and has no units.

Trigonometric Functions (or ratios)

• In a right-angle triangle ABC, let  ABC = Ѳ

Let AB = x, BC = y and AC = r.
Then we define the trigonometric ratios as under

Sign of trigonometric functions in various quadrants

• In quadrant I, all the trigonometric functions are positive.
• In quadrant II, only sine and cosec functions are positive.
• In quadrant III, only tan and cot functions are positive.
• In quadrant IV, only cosine and sec functions are positive.
• This is depicted as follows:

• In quadrants, where the y-axis is positive (i.e. I and II), sine is positive, and in quadrants where the x-axis is positive (i.e. I and IV), cosine is positive.
• A simple rule to remember the sign of the trigonometrical ratios, in all the four quadrants, is the four letter phrase—All School To College.
• Relations between Trigonometric Ratios
• Reciprocal relation:

• Quotient relation:
• Definition of Trigonometric Identity
An equation involving trigonometric ratios of an angle, say Ѳ, is termed as a trigonometric identity if it is satisfied by all values of Ѳ
Trigonometric Identities:
• sin2 Ѳ +cos2 Ѳ =1 ⟹ 1 – sin2 Ѳ = cos2 Ѳ and 1 – cos2 Ѳ = sin2 Ѳ
• 1=tan2 Ѳ = sec2 Ѳ ⟹ sec2 Ѳ –tan2 Ѳ =1 and sec2 Ѳ-1 = tan2 Ѳ
• 1+cot2 Ѳ = cosec2 Ѳ ⟹ cosec2 Ѳ – cot2 Ѳ =1 and cosec2 Ѳ- 1 cot
• Values of Trigonometrical Ratios of some Standard Angle

• Trigonometric Ratios of Complementary angles:
sin (90° – Ѳ) = cos Ѳ
cos (90°  – Ѳ)= sin Ѳ
tan (90°  – Ѳ) cot Ѳ
cot (90°  – Ѳ) tan Ѳ
sec (90°  – Ѳ) cosec Ѳ
cosec (90° – Ѳ) = sec Ѳ