θ and φ are the two acute angles of a right-angled triangle, and sin θ = 21/29. What is tan φ, as a fraction in its lowest terms? Answer in simplest form

20/21. Draw it: θ's opposite side 21, hypotenuse 29, adjacent √(29^2 − 21^2) = 20. φ's opposite side is θ's adjacent: tan φ = 20/21.

Trigonometry · Ages 14–15

Sine, cosine and tangent

sin = opp ÷ hyp, cos = adj ÷ hyp, tan = opp ÷ adj.

How to do it

Method

sin = opp ÷ hyp, cos = adj ÷ hyp, tan = opp ÷ adj

3, 4, 5: sin θ = 35, tan θ = 34

Another way

Label opp, adj, hyp first

SOH-CAH-TOA

In full

In a right-angled triangle, name the sides from the angle θ first: the hypotenuse is the longest, opposite the right angle; the opposite side faces θ; the adjacent side runs beside θ and isn't the hypotenuse. sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. Sides 3, 4, 5 with θ facing 3: sin θ = 35, cos θ = 45, tan θ = 34. An angle's ratios don't depend on the triangle's size: sides 6, 8, 10 give sin θ = 610 = 35 too, the triangles being similar. SOH-CAH-TOA helps (Sine opposite over hypotenuse, Cosine adjacent over hypotenuse, Tangent opposite over adjacent). Another way: write 'opp', 'adj' and 'hyp' on the sides in the picture, then match the formula. A calculator gives sin 35° and the like directly: check it is set to degrees (DEG).

Worked examples

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

Example 1
The right-angled triangle in the picture has sides of 12 cm, 5 cm and 13 cm. What is sin θ?
- A. 513
- B. 512
- 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.
Example 2
The right-angled triangle in the picture has a hypotenuse of 13 cm, and θ's opposite side is 5 cm. What is cos θ, as a fraction in its lowest terms?
Answer in simplest form
Show the answer and working
Answer: 1213
The adjacent side = 13² − 5² = 12 (Pythagoras); cos θ = 1213 = 1213.
Example 3
θ and φ are the two acute angles of a right-angled triangle, and sin θ = 2129. What is tan φ, as a fraction in its lowest terms?
Answer in simplest form
Show the answer and working
Answer: 2021
Draw it: θ's opposite side 21, hypotenuse 29, adjacent 29² − 21² = 20. φ's opposite side is θ's adjacent: tan φ = 2021.

Practice questions to try

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

1.The right-angled triangle in the picture has sides of 4 cm, 3 cm and 5 cm. What is sin θ?
- A. 45
- B. 34
- C. 35
- D. 43
Choose the right answer
Answer:
2.The right-angled triangle in the picture has sides of 12 cm, 5 cm and 13 cm. What is tan θ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
3.Find cos 19° on a calculator, correct to 4 decimal places: cos 19° ≈
Answer:
4.The right-angled triangle in the picture has a hypotenuse of 87 cm, and θ's opposite side is 60 cm. What is cos θ, as a fraction in its lowest terms?
Answer in simplest form
Answer:
5.In a right-angled triangle an acute angle θ has tan θ = 125, and the hypotenuse is 78 cm. How long is the side opposite θ, in cm?
Answer:

All the answers

1. C. 35
2. 512
3. 0.9455
4. 2129
5. 72 cm

Free worksheet: Sine, cosine and tangentWorking 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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