Without using a calculator, work out the exact value of 2\cos(60°) + \tan(45°).
Worked Solution & Answer
2(12) + 1 = 2 B1
Answer: 2
Exact trig. values — Notes & Worked Examples Show / Hide
Explanation:
For non-calculator exams, recall the exact values of sin θ, cos θ, and tan θ for 0°, 30°, 45°, 60°, and 90°. Derive them anytime from an isosceles right-angled triangle (legs 1, 1, hypotenuse √2) and an equilateral triangle of side 2 split in half (sides 1, √3, 2).
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 12 | √22 | √32 | 1 |
| cos θ | 1 | √32 | √22 | 12 | 0 |
| tan θ | 0 | √33 | 1 | √3 | — |
Example 1: Work out the exact value of sin 30° + cos 60° + tan 45°.
Solution:
Substitute: 12 + 12 + 1 = 2
Answer: 2
Example 2: Find the exact side x opposite 60° in a right-angled triangle with hypotenuse 12 cm.
Solution:
x = 12 × sin 60° = 12 × √32 = 6√3 cm
Answer: 6√3 cm
Without using a calculator, work out the exact value of \sin(30°) + \cos(60°).
Worked Solution & Answer
12 + 12 = 1 B1
Answer: 1
Find the exact value of \tan 30° × \sin 60°
Give your answer in its simplest form.
Give your answer in its simplest form.
Worked Solution & Answer
tan 30° = 1/√3, sin 60° = √3/2 M1
⇒ (1/√3) × (√3/2) = 1/2 A1
⇒ (1/√3) × (√3/2) = 1/2 A1
Answer: 12
Work out
(a) Circle the value of \sin 30°.
12 √(3)2 13 √(3)3 14
(b) Here is a right-angled triangle.
(a) Circle the value of \sin 30°.
12 √(3)2 13 √(3)3 14
(b) Here is a right-angled triangle.
(a) Circle the value of \sin 30°.
12 √(3)2 13 √(3)3 14 [1 mark]
12 √(3)2 13 √(3)3 14 [1 mark]
(b) Here is a right-angled triangle. [2 marks]
Work out the value of x.
x =
Worked Solution & Answer
(a)
sin 30° = 1/2. B1
Answer (a): 1/2
(b)
sin 30° = x / 24 M1
⇒ x = 24 × sin 30° = 24 × 1/2 = 12. A1
⇒ x = 24 × sin 30° = 24 × 1/2 = 12. A1
Answer (b): x = 12
(a) Write down the exact value of tan 45° (1) (b) Right-angled triangle with adjacent 4 cm, angle 60°, hypotenuse x cm. cos 60° = 0.5. Work out x. (2)
(a) Write down the exact value of \tan 45° [1 mark]
(b) Work out the value of x. [2 marks]
Worked Solution & Answer
(a)
tan 45° = 1 B1
Answer (a): 1
(b)
cos 60° = 4 / x M1
⇒ 0.5 = 4 / x
⇒ x = 8 A1
⇒ 0.5 = 4 / x
⇒ x = 8 A1
Answer (b): 8
(a) Write down the exact value of cos 30° (1) (b) Given that sin 30° = 0.5, work out the value of x in the right-angled triangle with hypotenuse 12 cm. (2)
(a) Write down the exact value of \cos 30° [1 mark]
(b) Given that \sin 30° = 0.5,
work out the value of x. [2 marks]
work out the value of x. [2 marks]
Worked Solution & Answer
(a)
√(3)2
Answer (a): √(3)2
(b)
x / 12 = sin 30° = 0.5 M1
⇒ x = 12 × 0.5 = 6 A1
⇒ x = 12 × 0.5 = 6 A1
Answer (b): 6
ABC is an equilateral triangle of side length 2 cm.
M is the midpoint of AC.
M is the midpoint of AC.
Using this diagram, show that \tan 30° = 1√(3).
Worked Solution & Answer
In right-angled triangle MBC: MC = 1 cm, ∠MBC = 30°, BM = √(2² − 1²) = √3 cm. M3
tan 30° = MC / BM = 1 / √3. C1
tan 30° = MC / BM = 1 / √3. C1
Answer: MC = 1 cm and angle MBC = 60° ÷ 2 = 30°. By Pythagoras in triangle MBC, BM = √(2² - 1²) = √3 cm, so tan 30° = opposite / adjacent = MC / BM = 1 / √3.
The table shows some values of x and y that satisfy the equation y = a \cos x° + b
Find the value of y when x = 45
Worked Solution & Answer
At x = 90: a cos 90° + b = 1 M2
⇒ b = 1. At x = 0: a cos 0° + 1 = 3 M1
⇒ a = 2. When x = 45: y = 2 cos 45° + 1 = 2(√2 / 2) + 1 = 1 + √2. A1
⇒ b = 1. At x = 0: a cos 0° + 1 = 3 M1
⇒ a = 2. When x = 45: y = 2 cos 45° + 1 = 2(√2 / 2) + 1 = 1 + √2. A1
Answer: 1 + √(2)
4 × \sin 30° × \tan 30° × \cos 30° = \sin y
Work out **one** possible value of y.
You **must** show your working.
Work out **one** possible value of y.
You **must** show your working.
Answer
degrees
Worked Solution & Answer
sin 30° = 1/2, tan 30° = 1/√3, cos 30° = √3/2. M2
4 × 1/2 × 1/√3 × √3/2 = 4 × 1/4 = 1. M1
sin y = 1 ⇒ y = 90°. A1
4 × 1/2 × 1/√3 × √3/2 = 4 × 1/4 = 1. M1
sin y = 1 ⇒ y = 90°. A1
Answer: Answer 90 degrees
Show that the value of 5\sin 30° × \cos 30° × 8\tan 30° is an integer.
Worked Solution & Answer
Exact values: M1
\sin 30° = 12
\cos 30° = √(3)2 M1
\tan 30° = 1√(3) (or √(3)3)
Substitute: M1
5 × 12 × √(3)2 × 8 × 1√(3)
= 5 × 14 × 8 × (√(3)√(3)) = 5 × 2 × 1 = 10
Since 10 is an integer, this proves the statement. C1
\sin 30° = 12
\cos 30° = √(3)2 M1
\tan 30° = 1√(3) (or √(3)3)
Substitute: M1
5 × 12 × √(3)2 × 8 × 1√(3)
= 5 × 14 × 8 × (√(3)√(3)) = 5 × 2 × 1 = 10
Since 10 is an integer, this proves the statement. C1
Answer: 10