Trigonometric Identities and graphs of inverse trigonometric ratios Pythagoras Theorem

Example

If asinѳ + bcosѳ = p

andacosѳ – bsinѳ = q

show that a² + b² = p² + q²

Solution

asinѳ + bcosѳ = p                                                        …(1)

acosѳ – bsinѳ = q                                                        …(2)

Squaring both sides of (1)

            (asinѳ + bcos)² = p²

: a² sin²ѳ + 2absinѳ cosѳ + b²cos²ѳ = p²                     …(3)

Squaring both sides of (2)

(acosѳ – bsinѳ)² = q²

a²cos²ѳ – 2absinѳ cosѳ + b²cos²ѳ = q²            …(4)

Adding (3) and (4)

a²sin²ѳ + 2absinѳ cosѳ + b²cos²ѳ + a²cosѳ – 2absinѳ cosѳ + b²sin²ѳ = p²q²

: a²sin²ѳ + a²cos²ѳ + b²cos²ѳ + b²sin²ѳ = p² + q²

a²(sin²ѳ + cos²ѳ) + b²(cos²ѳ + sin²ѳ) = p² + q²

But sin²ѳ + cos²ѳ = 1

: a² + b² = p² + q²

Evaluation

Sketch the graph of:

(i) y = sin2x                                                                 (ii) y = cos x

(iii) y = sec x                                                                (iv) cosec x

all at intervals of 30◦ range 0≤ x ≤ 360.

General Evaluation

(1) Draw the graph of y = 2cosx – 1 in the range 0◦ ≤ x ≤ 360◦ at intervals of 30◦.

(2) Draw the graph of y = 3sin x – 1 in the range of 0◦ ≤ x ≤ 360◦ at intervals of 30◦

(3) Prove that sec²ѳ + cosec²ѳ = (tanѳ + cotѳ) ².