Trigonometry

You will be able to

• Define sec θ, cosec θ and cot θ as the reciprocals of cos θ, sin θ and tan θ, and state where each one is undefined.
• Determine the sign of each trigonometric function in any quadrant using the CAST diagram, for positive and negative angles in degrees or radians.
• Find exact values such as sec 300° = 2 by using the reference angle, the quadrant sign and the special angles 30°, 45° and 60°.

You will be able to

• State the amplitude a and the period 360° ÷ b (2π ÷ b in radians) of y = a sin bx + c and y = a cos bx + c, and the period 180° ÷ b of y = a tan bx + c.
• Calculate the maximum c + a and the minimum c − a of a sine or cosine graph, and choose a and c to give a stated maximum and minimum.
• Sketch y = a tan bx + c on a given domain, marking its asymptotes at x = 90° ÷ b plus multiples of 180° ÷ b.

You will be able to

• Solve equations of the form sin(bx + p) = k, cos(bx + p) = k and tan(bx + p) = k in a stated domain by first finding the range of the argument bx + p.
• Solve quadratic equations in sin x, cos x or tan x, rejecting a root outside −1 to 1 with its reason.
• Use an identity such as sin²x + cos²x = 1 to write an equation in one trigonometric function, then solve it in degrees or in radians.

You will be able to

• Explain sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ from Pythagoras' theorem on the unit circle.
• Prove an identity by working from one side to the other, giving the identity or the algebra used on each line.
• Deduce that a statement such as sin 2x = 2 sin x is not an identity by finding a value of x for which the two sides differ.