Trigonometry

Skills and methods in this topic, easy to hard

1. 1Sine, cosine and tangentsin = opp ÷ hyp, cos = adj ÷ hyp, tan = opp ÷ adj

   Method3, 4, 5: sin θ = 35, tan θ = 34

   Another way: Label opp, adj, hyp firstSOH-CAH-TOA

2. 2Finding a sideUnknown on top: multiply; underneath: divide

   Methodx = 10 × sin 35° ≈ 5.7 cm

   Another way: Use the other acute angle10 × cos 55° ≈ 5.7

3. 3Finding an angleThe ratio, then sin⁻¹, cos⁻¹ or tan⁻¹

   Methodtan θ = 58, θ ≈ 32.0°

   Another way: 90° less the other angle90° − 58.0° = 32.0°

4. 4How the ratios are relatedsin² + cos² = 1; tan = sin ÷ cos; sin(90° − θ) = cos θ

   Methodsin θ = 513: cos θ = 1213, tan θ = 512

   Another way: Draw a triangle13² − 5² = 12

5. 5Exact values at 30°, 45° and 60°sin 30° = 12, sin 45° = 22, sin 60° = 32

   Method2 sin 30° + tan 45° = 1 + 1 = 2

   Another way: √1/2, √2/2, √3/2cos 60° = sin 30° = 12

6. 6Trigonometry in plane figuresFind the right-angled triangles; the shared side first

   MethodBC = htan B + htan C

   Another way: The shared side firstAD = AB sin B, then DC = AD ÷ tan C

7. 7Gradient and inclinationGradient = vertical ÷ horizontal = tan θ

   Method1 : 8: tan⁻¹(18) ≈ 7.1°

   Another way: Legs 1 and n1 + 8² = 65

8. 8Angles of elevation and depressionBoth from the horizontal; they are alternate angles

   Method50 × tan 32° ≈ 31.2 m

   Another way: The shared side firstThe sight line first, then the height

9. 9BearingsTrue bearings: from north, clockwise, three digits; back ± 180°

   MethodS40°E = 140°; the way back 320°

   Another way: Split into east and northd sin θ, d cos θ

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Sine, cosine and tangent →

   The right-angled triangle in the picture has sides of 5 cm, 12 cm and 13 cm. What is sin θ?

   • A. 512
   • B. 513
   • C. 125
   • D. 1213

   Choose the right answer

   ▶Show the answer and working

   Answer: D. 1213

   The hypotenuse is 13 (the longest, facing the right angle); sin θ = θ's opposite side ÷ 13.

2. Example 2Exact values at 30°, 45° and 60° →

   Without a calculator: 9 tan 45° − 3 cos 60° =

   ▶Show the answer and working

   Answer: 7.5

   tan 45° = 1, cos 60° = 12: 9 − 3 × 12 = 7.5.

3. Example 3Bearings →

   B is on a true bearing of 025° from A. As a compass bearing, what is B's bearing from A?

   • A. S25°E
   • B. N25°W
   • C. N25°E
   • D. S25°W

   Choose the right answer

   ▶Show the answer and working

   Answer: C. N25°E

   See which quarter 025° is in, then the angle from north or south, 25°.