Free Grade 3 Trigonometric Identities Worksheets
Free Grade 3 worksheets for Trigonometric Identities: 12 real questions with a full answer key and worked solutions. In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). Trigonometry links the sides to the angles.
Free
Higher
30 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- RememberIdentify trigonometric identities
Identify trigonometric identities accurately in a familiar context.
Assessed by: Exit ticket of four short items
- UnderstandExplain trigonometric identities
Explain trigonometric identities accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate trigonometric identities
Calculate trigonometric identities accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare trigonometric identities
Compare trigonometric identities accurately in a familiar context.
Assessed by: Exit ticket of four short items
Prerequisites
Close these gaps first — each has its own free lesson, worksheet and quiz.
How Trigonometric Identities works
In a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). Trigonometry links the sides to the angles.
12 practice questions
Easy to hard, 24 marks in total. Try them first, then open the answers.
- 1.A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.[1]
- 2.A right-angled triangle has shorter sides 5 cm and 12 cm. Find the hypotenuse.[1]
- 3.A right-angled triangle has shorter sides 8 cm and 15 cm. Find the hypotenuse.[1]
- 4.A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.[1]
- 5.A right-angled triangle has hypotenuse 15 cm and one side 9 cm. Find the other side.[2]
- 6.A right-angled triangle has hypotenuse 5 cm and one side 3 cm. Find the other side.[2]
- 7.A right-angled triangle has hypotenuse 25 cm and one side 7 cm. Find the other side.[2]
- 8.A right-angled triangle has hypotenuse 10 cm and one side 6 cm. Find the other side.[2]
- 9.In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 25 cm. Find θ to 1 d.p.[3]
- 10.In a right-angled triangle, the side opposite angle θ is 8 cm and the hypotenuse is 17 cm. Find θ to 1 d.p.[3]
- 11.In a right-angled triangle, the side opposite angle θ is 3 cm and the hypotenuse is 5 cm. Find θ to 1 d.p.[3]
- 12.In a right-angled triangle, the side opposite angle θ is 9 cm and the hypotenuse is 15 cm. Find θ to 1 d.p.[3]
Show answers and working
1. 10 cm
c² = a² + b² = 36 + 64 = 100. c = √100 = 10 cm.
2. 13 cm
c² = a² + b² = 25 + 144 = 169. c = √169 = 13 cm.
3. 17 cm
c² = a² + b² = 64 + 225 = 289. c = √289 = 17 cm.
4. 15 cm
c² = a² + b² = 81 + 144 = 225. c = √225 = 15 cm.
5. 12 cm
b² = c² − a² = 225 − 81 = 144. b = √144 = 12 cm.
6. 4 cm
b² = c² − a² = 25 − 9 = 16. b = √16 = 4 cm.
7. 24 cm
b² = c² − a² = 625 − 49 = 576. b = √576 = 24 cm.
8. 8 cm
b² = c² − a² = 100 − 36 = 64. b = √64 = 8 cm.
9. 16.3°
sin θ = opposite / hypotenuse = 7/25. θ = sin⁻¹(0.28) = 16.3°.
10. 28.1°
sin θ = opposite / hypotenuse = 8/17. θ = sin⁻¹(0.4706) = 28.1°.
11. 36.9°
sin θ = opposite / hypotenuse = 3/5. θ = sin⁻¹(0.6) = 36.9°.
12. 36.9°
sin θ = opposite / hypotenuse = 9/15. θ = sin⁻¹(0.6) = 36.9°.
Worked examples
A right-angled triangle has shorter sides 7 cm and 24 cm. Find the hypotenuse.
- c² = a² + b² = 49 + 576 = 625.
- c = √625 = 25 cm.
- Answer: 25 cm
A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side.
- b² = c² − a² = 169 − 25 = 144.
- b = √144 = 12 cm.
- Answer: 12 cm
In a right-angled triangle, the side opposite angle θ is 5 cm and the hypotenuse is 13 cm. Find θ to 1 d.p.
- sin θ = opposite / hypotenuse = 5/13.
- θ = sin⁻¹(0.3846) = 22.6°.
- Answer: 22.6°
Interactive practice
Free, no account required — answer on screen or print it.
Common mistakes
What learners typically get wrong, and how to fix it.
Adds the squares when finding a shorter side.
Correction: Shorter side: subtract from the hypotenuse squared.
Forgets to square root at the end.
Correction: c² is not c — finish with √.
Teacher tips
- · Label the hypotenuse first every time.
Parent tips
- · Measure a phone screen's sides and check the diagonal.
Real-life applications
- · Ladder length against a wall, TV screen sizes.
Assessment objectives
How mastery is evidenced, and which standards it maps to.
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Frequently asked
- Is this Grade 3 Trigonometric Identities worksheet really free?
- Yes. Every worksheets on FreeMathWorksheet is free to use, print and share — no account, no paywall and no watermark.
- What should a learner already know before this?
- Start with Tangent Graph, Cosine Graph, Non-right-angled Trigonometry. Each one has its own free lesson, worksheet and quiz.
- How is the worksheet sequenced?
- Recognition first, then understanding, application, reasoning and finally mastery — the same ladder used across every free resource on the platform.
- What comes next after Trigonometric Identities?
- Move on to Right-angled Trigonometry, Non-right-angled Trigonometry, which build directly on this idea.
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