Question

Explain Trigonometry functions with table and give some common Trigonometry Formuals with 1 example only .

Answer 1

EXPLANATION.

Fundamental trigonometric identities.

(1) = sin²θ + cos²θ = 1.

(2) = 1 + tan²θ = sec²θ.

(3) = 1 + cot²θ = cosec²θ.

Sign of trigonometric ratios of function.

In first quadrant all trigonometric functions is (+ve).

In second quadrant sinθ and cosecθ are +ve.

In third quadrant tanθ and cotθ are +ve.

In fourth quadrant cosθ and secθ are +ve.

By Pythagoras theorem, we get.

⇒ P² + B² = H².

Hypotenuse > Perpendicular > Base.

(1) = sinθ = perpendicular/hypotenuse.

(2) = cosθ = base/hypotenuse.

(3) = tanθ = perpendicular/base.

(4) = cosecθ = hypotenuse/perpendicular.

(5) = secθ = hypotenuse/base.

(6) = cotθ = base/perpendicular.

(1) = secθ = 1/cosθ.

(2) = cosecθ = 1/sinθ.

(3) = tanθ = sinθ/cosθ.

(4) = cotθ = cosθ/sinθ.

(1) = sin(-θ) = - sinθ.

(2) = cos(-θ) = cosθ.

(3) = tan(-θ) = - tanθ.

(4) = cosec(-θ) = - cosecθ.

(5) = sec(-θ) = secθ.

(6) = cot(-θ) = - cotθ.

Sum and difference formula.

(1) = sin(A + B) = sin(A).cos(B) + cos(A).sin(B).

(2) = sin(A - B) = sin(A).cos(B) - cos(A).sin(B).

(3) = cos(A + B) = cos(A).cos(B) - sin(A).sin(B).

(4) = cos(A - B) = cos(A).cos(B) + sin(A).sin(B).

(5) = tan(A + B) = tan(A) + tan(B)/1 - tan(A).tan(B).

(6) = tan(A - B) = tan(A) - tan(B)/1 + tan(A).tan(B).

Trigonometrical ratios of multiple angles.

(1) = sin2θ = 2sinθ.cosθ = 2tanθ/1 + tan²θ.

(2) = cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ = 1 - tan²θ/1 + tan²θ.

(3) = tan2θ = 2tanθ/1 - tan²θ.

(4) = sin3θ = 3sinθ - 4sin³θ.

(5) = cos3θ = 4cos³θ - 3cosθ.

(6) = tan3θ = 3tanθ - tan³θ/1 - 3tan²θ.

(7) = sin(θ/2) = √1 - cosθ/2.

(8) = cos(θ/2) = √1 + cosθ/2.

(9) = tan(θ/2) = √1 - cosθ/1 + cosθ = 1 - cosθ/sinθ = sinθ/1 + cosθ.

Answer 2

Trigonometrical ratios :

• sin ∅ = P/H

• cos ∅ = B/H

• tan ∅ = P/B

• cot = B/P

• sec = H/B

• cosec = H/P

Here,

P refers Perpendicular or Height

B refers Base

H refers Hypotentuse

Reciprocal Relations :

$\sf \small(1) \: sin \: \theta = \frac{1}{cosec \: \theta}$

$\sf \small(2)cosec \: \theta = \frac{1}{sin \: \theta}$

$(3) cos \: \theta = \frac{1}{sec \: \theta}$

$\sf \small(4) sec \: \theta = \frac{1}{cos\: \theta}$

$\sf \small(5) tan \: \theta = \frac{1}{cot \: \theta}$

$\sf \small(6) cot \: \theta = \frac{1}{tan \: \theta}$

From the above relations it follows that :-

• sin ∅ × cosec ∅ = 1

• cos ∅ × sec ∅ = 1

• tan ∅ × cot ∅ = 1

Quotient Relations :

$\sf \small tan \: \theta = \frac{sin \: \theta}{cos \: \theta}$

$\sf \small cot \theta = \frac{cos \: \theta}{sin \: \theta}$

Square Relations :

• sin² ∅ + cos² ∅ = 1

• sec² ∅ – tan² ∅ = 1

• cosec² ∅ – cot² ∅ = 1

Remark : sin² ∅ means (sin ∅)² and sin² ∅ is read as sine squared ∅.

Similarly it is same for others too !!

Trigonometrical ratios of complementary angles :

• sin(90° – ∅) = cos ∅

• cos(90° – ∅) = sin ∅

• tan(90° – ∅) = cot ∅

• cot(90° – ∅) = tan ∅

• sec(90° – ∅) = cosec ∅

• cosec(90° – ∅) = sec ∅