Arc length & sector area

The diagram shows a triangle ABC. The arc BD is part of a circle with centre A and radius 6 cm.

The area of the sector ABD is 14.4 cm².

Show that angle BAD is 0.8 radians.

The area of the triangle ABC is three times the area of the sector ABD.

Find the length AC.

Find the perimeter of the region BCD.

The compound angle formulae for sin (A + B) and sin (A - B) are

sin (A + B) = sin A cos B + cos A sin B and sin (A - B) = sin A cos B - cos A sin B.

By letting C = A + B and D = A - B, show that

Hence write down a similar inequality interval for the expression

Using parts (a) and (c)(ii), show from first principles that the derivative of sin x is cos x,

where x is measured in radians.

In this question you must show detailed reasoning.

The diagram shows a sector AOB of a circle with centre O and radius (3 x + 1)cm. The angle AOB is 2 radians. The area of sector AOB is less than (44x - 7)cm².

Find the set of possible values of x. Give your answer in set notation.

Show that θ = 1.25 sin θ.

The equation θ = 1.25 sin θ has only one root for θ > 0.

This root can be found by using the iterative formula θ_n+1 = 1.25 sin θ_n with a starting value of θ₁ = 0.5.

• Write down the values of θ₂, θ₃ and θ₄.

• Hence find the value of this root correct to 3 significant figures.

Draw a suitable diagram to show why using an iterative process with the formula

θ_n+1 = sin^-1(0.8θ_n) does not converge to the root found in (b).

Show that θ = 0.4 + sin θ.

Let F (θ) = 0.4 + sin θ.

By considering the value of F ′ (θ) where θ = 1.2, explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that 1.2 is sufficiently close to

the root.

Use the iterative formula θ_n+1 = 0.4 + sin θ_n with a starting value of 1.2 to find the value of θ correct to 4 significant figures.

You should show the result of each iteration.

Use a change of sign method to show that the value of θ found in part (c) is correct to 4

significant figures.