Trigonometric Calculations

1. Define the trigonometric ratios using sinθ, cos θ and tan θ, using right angles triangles.

2. Extend the definitions for sinθ, cos θ and tan θ for 0°< θ<360°

3. Define the reciprocals of trigonometric ratios cosec θ, sec θ, cot θ

4. Derive values of trigonometric ratios for special cases (without using a calculator) θЄ{0 °; 30 °; 45 °; 60 °; 90 °}

5. Solve two dimensional problems involving right angled triangles.

6. Solve simple trigonometric equations for acute angles.

7. Use diagrams to determine the numerical values of ratios for angles from 0 ° to 360 °.

Calculator Usage

ExamplesDetermine the value of each of the following using a calculator (Round off your answer to twp decimal places).

A) sin73°

B) 2sin20 ° + 3cos10 °

C) tan(21 ° + 36 °)

continuedSolutions

A) sin73°=0,9563048≈ 0,96

C) tan(21 ° + 36 °)= tan 57°= 1,539865 ≈ 1,54

B) 2sin20 °+ 3cos10 °= 0,6840403 + 2,9544233= 3,6384638 ≈ 3,64

Calculator usageTo determine the angle if the ratio is given, is made possible on a

calculator if you press the shift key or second function key, then the trigonometric function key.

The keys sin-1,cos-1and tan-1 is used to show the inverse function.

Solve trigonometric equations

To solve for θ in the following problem if θ is an acute angle.Tanθ = Key in the following on your calculator. 11.9; = ; shift ; tan-1; ans or =

Therefore θ = 18,1°

5,36

9,11

Trigonometric EquationsExamples

1) Solve for θ (Round off your answer to ONE decimal place) (θ is acute)

a) sin θ = 0,823b) 2cos θ = 1,264c) cos2 θ = 0,943d) tan(θ - 25°) = 0,465e) sin2 θ = 0,326

2

Solutionsa) sin θ = 0,823

θ = 55,4 °

b) 2cos θ = 1,264 cos θ = 0,632

θ = 50,8 °

c) cos2 θ = 0,943 2θ = 19,438371 ° θ = 9,711855° θ = 9,7°

continuedd) tan(θ - 25°) = 0,465 θ - 25°= 24,938427°

θ = 49,938427° θ = 49,9°

e) sin2 θ = 0,326 2

sin2 θ = 0,652 2 θ = 40,69256357°

θ = 20,34628179° θ = 20,3°

Test your Knowledge

Question Solve for θ if θ ε [ 0° ; 90°] and 2 tan θ = 1 Answer A 31.6° B 26.6° C 45.1° D 13.6°

Test your knowledge

Question Solve for θ if θ ε [ 0° ; 90°] and ⅓ sin θ = 0.2 Answer A 31.6° B 36.9° C 23.1° D 63.6°

Test your knowledge

Question Solve for θ if θ ε [ 0° ; 90°] and

23 cos θ - 1 = 0

Answer A 23,9° B 48.2° C 21.8° D 56.4°