Trigonometry (I) · Ages 13–14
How the ratios change with the angle
θ up: sin and tan up, cos down; sin and cos between 0 and 1.
How to do it
Method
θ up: sin and tan up, cos down; sin and cos between 0 and 1
cos 20° > cos 55° > cos 80°
Another way
Each value on a calculator
0.940 > 0.574 > 0.174
In full
Fix a right-angled triangle's hypotenuse at 1 and let the acute angle θ grow from 0°: the opposite side gets longer and the adjacent side shorter, so sin θ rises with θ, cos θ falls, and tan θ = opposite ÷ adjacent rises. The hypotenuse is the longest side, so an acute angle's sin θ and cos θ always lie between 0 and 1 — never 1.2; tan θ can be more than 1 (for θ over 45°). You can compare without a calculator: cos 20° > cos 55° > cos 80°, as the bigger the angle, the smaller its cosine. Careful: a ratio can't be split up. sin 20° + sin 40° ≈ 0.342 + 0.643 = 0.985, but sin 60° ≈ 0.866, so sin 20° + sin 40° ≠ sin 60°; nor is 2 sin 30° the same as sin 60°. Another way: find each value on a calculator and compare, cos 20° ≈ 0.940, cos 55° ≈ 0.574, cos 80° ≈ 0.174.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
θ is an acute angle. As θ grows from 22° to 53°, what happens to cos θ?
- A. Up first, then down
- B. It gets bigger
- C. It gets smaller
- D. It stays the same
Choose the right answer
Show the answer and working
Answer: C. It gets smaller
Fix the hypotenuse and let θ grow from 22° to 53°: the adjacent side gets shorter and shorter, so cos θ = adjacent ÷ hypotenuse keeps falling.
Example 2
With a calculator, correct to 4 decimal places: sin 36° + sin 15° ≈ and sin 51° ≈ .
Fill in each blank
Show the answer and working
Answer: 0.8466, 0.7771
sin 36° + sin 15° ≈ 0.8466 and sin 51° ≈ 0.7771: not the same. A ratio can't be split up: sin(36° + 15°) ≠ sin 36° + sin 15°.
Example 3
θ is an acute angle with cos θ = 11n, n a positive whole number no more than 35. How many different values can n take?
Show the answer and working
Answer: 24
An acute angle's cosine is between 0 and 1: 11n < 1, so n must be over 11; it can be 11 + 1 up to 35, 35 − 11 values.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.There can be an acute angle θ with cos θ = 0.5.
True or false?
Answer:
2.No calculator needed: sin 18° is bigger than sin 71°.
True or false?
Answer:
3.Without a calculator, which of these is the biggest?
- A. cos 83°
- B. cos 30°
- C. cos 11°
- D. cos 47°
Choose the right answer
Answer:
4.θ is an acute angle with tan θ = 0.9, so θ is more than 45°.
True or false?
Answer:
5.Without a calculator: for θ = 57°, how do sin θ, cos θ and tan θ go, biggest first?
- A. tan θ > sin θ > cos θ
- B. cos θ > sin θ > tan θ
- C. sin θ > tan θ > cos θ
- D. sin θ > cos θ > tan θ
Choose the right answer
Answer:
All the answers
- 1. True
- 2. False
- 3. C. cos 11°
- 4. False
- 5. A. tan θ > sin θ > cos θ
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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