The exact value of sin 30° is ½ and sin²θ + cos²θ equals 1 for any angle θ, so this Year 12 sheet opens with an exact value and the Pythagorean identity. The answers record 30° and 150° together with 360° as a period.
Key Takeaways
sin 30° = ½ (Q1).
sin²θ + cos²θ = 1 for any angle θ (Q2).
Q6 records 360° as a period.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.The exact value of sin 30° is:
(a) ½(b) √3 ÷ 2(c) 1(d) √2 ÷ 2
2.For any angle θ, the value of sin²θ + cos²θ is:
(a) 1(b) 0(c) 2(d) tan θ
3.The solutions of sin θ = ½ for 0° ≤ θ < 360° are:
(a) 30° and 150°(b) 30° only(c) 150° and 210°(d) 60° and 120°
4.The exact value of tan 45° is:
(a) 1(b) 0(c) ½(d) √3
5.The exact value of cos 60° is:
(a) ½(b) √3 ÷ 2(c) 1(d) √2 ÷ 2
6.The period of the function y = sin x is:
(a) 360°(b) 180°(c) 90°(d) 45°
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Solve cos θ = √3 ÷ 2 for 0° ≤ θ ≤ 360°.
8.Solve 2 sin θ = √3 for 0° < θ < 360°.
9.Given tan θ = ¾ with θ acute, find the exact values of sin θ and cos θ.
10.Solve tan θ = 1 for 0° ≤ θ ≤ 360°.
11.Solve sin θ = −½ for 0° < θ < 360°.
12.Given sin θ = 5 ÷ 13 with θ acute, find the exact value of cos θ.
13.Solve 2 cos θ − 1 = 0 for 0° ≤ θ ≤ 360°.
14.Show that tan θ sin θ = (1 − cos²θ) ÷ cos θ for all angles where cos θ is not zero.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.From a point on level ground, the angle of elevation of the top of a tree is 30°. The point is 20 m from the base of the tree. Find the height of the tree in terms of √3.
16.A kite is flying on a taut string 30 m long held at an angle of 30° above the horizontal. Find the height of the kite above the ground.
17.The height in metres of a tide above mean level is modelled by h = 4 sin(30t°), where t is the time in hours. Find the height when t = 1 and state the maximum height of the tide.
18.A boat is 50 m from the base of a cliff and the angle of elevation of the top of the cliff is 45°. Find the height of the cliff.
19.An isosceles triangle has equal sides of length 10 cm and a base of length 12 cm. Find the angle at the apex, giving your answer to the nearest degree.
20.A ladder of length 5 m leans against a wall making an angle of 60° with the ground. Find the distance from the foot of the ladder to the wall.
Answer Key
1. a) ½
2. a) 1
3. a) 30° and 150°
4. a) 1
5. a) ½
6. a) 360°
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the exact value of sin 30°?
½, option (a) in this worksheet (Q1).
What does sin²θ + cos²θ equal?
1, option (a) in this worksheet (Q2).
Which angles satisfy sin θ = ½ in the first two quadrants?
30° and 150°, option (a) in this worksheet (Q3).
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