θ is acute and sin θ = 4/5. What is cos θ, as a fraction? Answer in simplest form

3/5. cos² θ = 1 − sin² θ = 1 − (4/5)^2, so cos θ = 3/5 (positive for an acute angle). Or draw the triangle: the adjacent side √(5^2 − 4^2) = 3.

There can be an acute angle θ with sin θ = 24/25 and cos θ = 7/25. True or false?

True. sin² θ + cos² θ must be 1: (24/25)^2 + (7/25)^2 = 1, so cos θ can only be 7/25.

θ is acute and tan θ = 15/8. As fractions: sin θ = □, cos θ = □. Fill in each blank

15/17, 8/17. Draw a right-angled triangle with opposite 15 and adjacent 8: the hypotenuse is √(15^2 + 8^2) = 17, so sin θ = 15/17 and cos θ = 8/17.

Trigonometry · Ages 14–15

How the ratios are related

sin² + cos² = 1; tan = sin ÷ cos; sin(90° − θ) = cos θ.

How to do it

Method

sin² + cos² = 1; tan = sin ÷ cos; sin(90° − θ) = cos θ

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

Another way

Draw a triangle

13² − 5² = 12

In full

An acute angle θ's ratios are tied together. tan θ = sin θ ÷ cos θ: (opp/hyp) ÷ (adj/hyp) = opp/adj. sin² θ + cos² θ = 1: by Pythagoras' theorem opp² + adj² = hyp², divided through by hyp². sin(90° − θ) = cos θ and cos(90° − θ) = sin θ: a right-angled triangle's two acute angles make 90°, and one's opposite side is the other's adjacent. Given sin θ = 513, for cos θ: cos² θ = 1 − 25169 = 144169, so cos θ = 1213 (an acute angle's ratios are positive), and tan θ = 512. Another way: draw a right-angled triangle with opposite 5 and hypotenuse 13, find the adjacent side by Pythagoras, 169 − 25 = 12, and read off cos θ = 1213 and tan θ = 512.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
θ is acute and sin θ = 45. What is cos θ, as a fraction?
Answer in simplest form
Show the answer and working
Answer: 35
cos² θ = 1 − sin² θ = 1 − (45)², so cos θ = 35 (positive for an acute angle). Or draw the triangle: the adjacent side 5² − 4² = 3.
Example 2
There can be an acute angle θ with sin θ = 2425 and cos θ = 725.
True or false?
Show the answer and working
Answer: True
sin² θ + cos² θ must be 1: (2425)² + (725)² = 1, so cos θ can only be 725.
Example 3
θ is acute and tan θ = 158. As fractions: sin θ = , cos θ = .
Fill in each blank
Show the answer and working
Answer: 1517, 817
Draw a right-angled triangle with opposite 15 and adjacent 8: the hypotenuse is 15² + 8² = 17, so sin θ = 1517 and cos θ = 817.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.θ is acute, with sin θ = 2425 and cos θ = 725. What is tan θ?
- A. 168625
- B. 247
- C. 257
- D. 724
Choose the right answer
Answer:
2.θ is acute and sin θ = 1517. What is cos θ, as a fraction?
Answer in simplest form
Answer:
3.θ is acute and cos θ = 35. What is 1 − sin² θ, as a fraction?
Answer in simplest form
Answer:
4.Given sin 64° ≈ 0.8988, without a calculator, cos 26° ≈ (4 decimal places)
Answer:
5.θ is acute and sin θ = 2129. What is tan θ + 1tan θ, as a fraction?
Answer in simplest form
Answer:

All the answers

1. B. 247
2. 817
3. 925
4. 0.8988
5. 841420

Free worksheet: How the ratios are relatedWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

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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