# 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:**

- sin
^{2}Ѳ +cos^{2}Ѳ =1 ⟹ 1 – sin^{2}Ѳ = cos^{2}Ѳ and 1 – cos^{2}Ѳ = sin^{2}Ѳ - 1=tan
^{2}Ѳ = sec^{2}Ѳ ⟹ sec2 Ѳ –tan2 Ѳ =1 and sec^{2}Ѳ-1 = tan^{2}Ѳ - 1+cot
^{2}Ѳ = cosec^{2}Ѳ ⟹ cosec^{2}Ѳ – cot^{2}Ѳ =1 and cosec^{2}Ѳ- 1 cot^{2 }

**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

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