Photo AI
1. (a) By writing sin 30° as sin (2θ + θ), show that sin 30° = 3sin θ - 4sin³ θ - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 6
Question 3
1. (a) By writing sin 30° as sin (2θ + θ), show that sin 30° = 3sin θ - 4sin³ θ. (b) Given that sin θ = \frac{\sqrt{3}}{4}, find the exact value of sin 30°.
Worked Solution & Example Answer:1. (a) By writing sin 30° as sin (2θ + θ), show that sin 30° = 3sin θ - 4sin³ θ - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 6
Step 1
(a) By writing sin 30° as sin (2θ + θ), show that
Answer
To show that ( \sin 30° = 3\sin \theta - 4\sin^3 \theta ), we can use the angle addition formula for sine:
[ \sin(2\theta + \theta) = \sin 2\theta \cos \theta + \cos 2\theta \sin \theta ]
First, we know from trigonometric identities:
- ( \sin 2\theta = 2\sin \theta \cos \theta )
- ( \cos 2\theta = 1 - 2\sin^2 \theta )
Substituting these into the equation gives us:
[ \sin(2\theta + \theta) = (2\sin \theta \cos \theta) \cos \theta + (1 - 2\sin^2 \theta) \sin \theta ]
This leads to:
[ = 2\sin \theta \cos^2 \theta + \sin \theta - 2\sin^3 \theta ]
Using ( \cos^2 \theta = 1 - \sin^2 \theta ), we can substitute again:
[ = 2\sin \theta (1 - \sin^2 \theta) + \sin \theta - 2\sin^3 \theta ]
Expanding yields:
[ = 2\sin \theta - 2\sin^3 \theta + \sin \theta - 2\sin^3 \theta = 3\sin \theta - 4\sin^3 \theta ]
Thus, we have shown that: [ \sin 30° = 3\sin \theta - 4\sin^3 \theta ]
Step 2
(b) Given that sin θ = \frac{\sqrt{3}}{4}, find the exact value of sin 30°.
Answer
Given ( \sin \theta = \frac{\sqrt{3}}{4} ), we can substitute this into the equation we derived in part (a):
[ \sin 30° = 3\left(\frac{\sqrt{3}}{4}\right) - 4\left(\frac{\sqrt{3}}{4}\right)^3 ]
Calculating the first term:
[ = \frac{3\sqrt{3}}{4} ]
For the second term:
[
- 4\cdot\frac{\sqrt{3}}{4}\cdot\frac{3\sqrt{3}}{16} = -\frac{3\sqrt{3}}{4} ]
Thus:
[ \sin 30° = \frac{3\sqrt{3}}{4} - \frac{3\sqrt{3}}{16} ]
Finding a common denominator:
[ = \frac{12\sqrt{3}}{16} - \frac{3\sqrt{3}}{16} = \frac{9\sqrt{3}}{16} ]
Therefore, the exact value of ( \sin 30° ) is ( \frac{9\sqrt{3}}{16} ).