A-Level Edexcel Maths Pure Proof: 1. (a) By writing sin 30° as sin

1. (a) By writing sin 30° as sin (2θ + θ), show that sin 30° = 3sin θ - 4sin³ θ - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 6

Question 2

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 2 - 2007 - Paper 6

Step 1

By writing sin 30° as sin (2θ + θ), show that

96%

114 rated

Answer

To show that sin 30° can be expressed as 3sin θ - 4sin³ θ, we start by using the angle addition formula:

sin(2 heta + heta) = sin 2 heta \cdot cos \theta + cos 2 heta \cdot sin \theta

Where:

We know that ( sin 2 heta = 2sin \theta \cdot cos \theta ) and ( cos 2 heta = 1 - 2sin^2 \theta ).

Substituting these into the equation gives:

sin(2 heta + \theta) = (2sin \theta \cdot cos \theta) \cdot cos \theta + (1 - 2sin^2 \theta) \cdot sin \theta

This simplifies to:

= 2sin \theta \cdot cos^2 \theta + sin \theta - 2sin^3 \theta

Now, recognizing that ( cos^2 \theta = 1 - sin^2 \theta ) leads to:

= 2sin \theta \cdot (1 - sin^2 \theta) + sin \theta - 2sin^3 \theta

Expanding this results in:

= 2sin \theta - 2sin^3 \theta + sin \theta - 2sin^3 \theta

Combining like terms gives:

sin 30° = 3sin \theta - 4sin^3 \theta

This concludes the proof.

Step 2

Given that sin θ = \( \frac{\sqrt{3}}{4} \), find the exact value of sin 30°.

99%

104 rated

Answer

Given that ( sin \theta = \frac{\sqrt{3}}{4} ), we can substitute this value into our previously derived equation:

sin 30° = 3 \cdot \frac{\sqrt{3}}{4} - 4 \left(\frac{\sqrt{3}}{4}\right)^3

Calculating the first term:

= \frac{3\sqrt{3}}{4}

For the second term, we first evaluate ( \left(\frac{\sqrt{3}}{4}\right)^3 ):

= \frac{3\sqrt{3}}{64}

Now substituting back gives us:

sin 30° = \frac{3\sqrt{3}}{4} - 4 \cdot \frac{3\sqrt{3}}{64}

Simplifying the second term:

= \frac{3\sqrt{3}}{4} - \frac{3\sqrt{3}}{16}

To combine these fractions, we convert ( \frac{3\sqrt{3}}{4} ) to a fraction with the same denominator:

= \frac{12\sqrt{3}}{16} - \frac{3\sqrt{3}}{16}

Finally:

sin 30° = \frac{9\sqrt{3}}{16}

This is the exact value of sin 30°.

1. (a) By writing sin 30° as sin (2θ + θ), show that sin 30° = 3sin θ - 4sin³ θ - Edexcel - A-Level Maths Pure - Question 2 - 2007 - Paper 6

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 2 - 2007 - Paper 6

By writing sin 30° as sin (2θ + θ), show that

Given that sin θ = \( \frac{\sqrt{3}}{4} \), find the exact value of sin 30°.

Join the A-Level students using SimpleStudy...