Compound & double angle formulae

Prove that (cos θ + sin θ)² ≡ 1 + sin 2θ.

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.

Use the result cos (A + B) = cos A cos B - sin A sin B to show that

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

The function f (θ) is defined as cos (θ + 30 °) cos (θ - 30 °), where θ is in degrees.

Determine the following.

• The maximum value of f (θ)

• The smallest positive value of θ for which this maximum value occurs

Determine the following.

• The minimum value of f (θ)

• The smallest positive value of θ for which this minimum value occurs

The diagram shows part of the graph of y = x². The normal to the curve at the point A(1, 1) meets the curve again at B. Angle AOB is denoted by α.

Determine the coordinates of B.

Hence determine the exact value of tan α.

The diagram shows triangle ABC. The perpendicular from C to AB meets AB at D.

Angle ACD = x, angle D C B = y, length BC = a and length AC = b.

Explain why the length of CD can be written as a cos y.

Show that the area of the triangle ADC is given by ab sin x cos y.

Hence, or otherwise, show that sin (x + y) = sin x cos y + cos x sin y.

Differentiate cos x with respect to x, from first principles.

Three forces, of magnitudes 4 N, 6 N and P N, act at a point in the directions shown in the diagram.

The forces are in equilibrium.

Show that θ = 41.4°, correct to 3 significant figures.

Hence find the value of P.

The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.

Find

the magnitude of the resultant of the two remaining forces,

the direction of the resultant of the two remaining forces.