Everything Maths Grade 12

10.4 CHAPTER 10. TRIGONOMETRY

Summary of the Trigonometric

Rules and Identities

EMCCT

Pythagorean Identity Cofunction Identities Ratio Identities

cos^2 θ + sin^2 θ = 1 sin(90◦−θ) = cosθ tanθ =cossin θ θ

cos(90◦−θ) = sinθ

Odd/Even Identities Periodicity Identities Double Angle Identities

sin(−θ) =− sinθ sin(θ± 360 ◦) = sinθ sin(2θ) = 2 sinθ cosθ

cos(−θ) = cosθ cos(θ± 360 ◦) = cosθ cos (2θ) = cos^2 θ− sin^2 θ

tan(−θ) =− tanθ tan(θ± 180 ◦) = tanθ cos (2θ) = 1− 2 sin^2 θ

tan (2θ) = 1 2tan−tan θ (^2) θ
Addition/Subtraction Identities Area Rule Cosine rule
sin (θ +φ) = sinθ cosφ + cosθ sinφ Area =^12 bc sinA a^2 = b^2 +c^2 − 2 bc cosA
sin (θ−φ) = sinθ cosφ− cosθ sinφ Area =^12 ab sinC b^2 = a^2 +c^2 − 2 ac cosB
cos (θ +φ) = cosθ cosφ− sinθ sinφ Area =^12 ac sinB c^2 = a^2 +b^2 − 2 ab cosC
cos (θ−φ) = cosθ cosφ + sinθ sinφ
tan (θ +φ) = 1 tan−tan φ+tan θtan θ φ
tan (θ−φ) =1+tantan φ− θtantan θ φ
Sine Rule
sin A
a =
sin B
b =
sin C
c
Chapter 10 End of Chapter Exercises
Do the following without using a calculator.

1. Suppose cosθ = 0, 7. Find cos 2θ and cos 4θ.


2. If sinθ =^47 , again find cos 2θ and cos 4θ.


3. Work out the following:
   (a) cos 15◦
   (b) cos 75◦
   (c) tan 105◦
   (d) cos 15◦
   (e) cos 3◦cos 42◦− sin 3◦sin 42◦
   (f) 1 − 2 sin^2 (22, 5 ◦)


4. Solve the following equations:
   (a) cos 3θ. cosθ− sin 3θ. sinθ =−^12