Bridge to university · Mathematics

Bridge: trigonometry for first-year courses

Radians, exact unit circle values, identities, solving trig equations and sinusoidal graphs, reviewed for students starting university math.

36 cards5 topicsAbout 15 minutes a roundGrade 12 trigonometry review before first-year calculus, physics and engineering

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About this deck

Trigonometry shows up everywhere in first year: derivatives of sine and cosine in calculus, vector components in physics, waves and oscillations in engineering. This deck reviews the pieces that instructors expect you to know without a calculator. You will see radian measure and conversions, exact values on the unit circle, the CAST rule and reference angles, the main identities (reciprocal, quotient, Pythagorean, sum and double angle), methods for solving trig equations on a given interval, and the features of sinusoidal and tangent graphs.

Use the deck if you are finishing Grade 12 math with a trigonometry unit, or if your trig is rusty after a gap year. Work in radians throughout, because university calculus does. For the exact value cards, sketch the 30-60-90 and 45-45-90 triangles from memory instead of memorizing a table; it is faster and harder to forget. When a card states an equation to solve, write the full solution set before flipping. Watch for the classic errors: leaving the calculator in degree mode, dividing both sides by sin x and losing solutions, treating sin(2x) as 2 sin x, and forgetting that an inverse trig key returns only one angle.

Before you start: Right-triangle trigonometry and basic function graphing from Grade 11.

All 36 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Radians

Radian: The angle at the centre of a circle that cuts off an arc as long as the radius. A full turn is 2pi of these units.Example: 180 degrees equals pi of them
Converting degrees to radians: Multiply the degree measure by pi/180.Example: 135 degrees = 135pi/180 = 3pi/4
Converting radians to degrees: Multiply the radian measure by 180/pi.Example: 5pi/6 = 150 degrees
Arc length s = r*theta: Length along a circle equals radius times angle, and it works only when the angle is measured in radians.Example: r = 4 cm and theta = pi/2 give 2pi cm, about 6.28 cm
Calculator mode: Set the calculator to radians for calculus and most university work. A degree setting is the most common cause of wrong decimals.Example: sin(1) is about 0.841 in radians but about 0.017 in degrees

Unit circle

Unit circle: The circle of radius 1 centred at the origin. The point at angle theta has coordinates (cos theta, sin theta).Example: At pi/2 the point is (0, 1)
sin(pi/6) and cos(pi/6): 1/2 and sqrt(3)/2, read from the 30-60-90 triangle with sides 1, sqrt(3) and 2.Example: pi/6 is 30 degrees
sin(pi/4) and cos(pi/4): Both equal sqrt(2)/2, about 0.707, from the isosceles right triangle with legs 1 and 1.Example: pi/4 is 45 degrees
sin(pi/3) and cos(pi/3): sqrt(3)/2 and 1/2; the two values swap compared with pi/6.Example: pi/3 is 60 degrees
tan at pi/6, pi/4 and pi/3: sqrt(3)/3 (the same as 1/sqrt(3)), then 1, then sqrt(3).Example: tan rises steeply as the angle nears pi/2
Quadrantal angles 0, pi/2, pi, 3pi/2: Points (1, 0), (0, 1), (-1, 0) and (0, -1), so cosine is the x coordinate and sine the y coordinate.Example: cos(pi) = -1 and sin(3pi/2) = -1
CAST rule: Shows which ratios are positive: quadrant I all, II sine, III tangent, IV cosine, counting counter-clockwise from the positive x axis.Example: cos(2pi/3) is negative because 2pi/3 lies in quadrant II
Reference angle: The acute angle between the terminal arm and the x axis. Trig values match it in size; the quadrant sets the sign.Example: For 5pi/4 it is pi/4, so sin(5pi/4) = -sqrt(2)/2
Coterminal angles: Angles that differ by a multiple of 2pi share a terminal arm, so all their trig values are identical.Example: pi/3 and 7pi/3

Identities

Reciprocal ratios: csc x = 1/sin x, sec x = 1/cos x and cot x = 1/tan x.Example: sec(pi/3) = 1/(1/2) = 2
Quotient identity: tan x = sin x / cos x, and cot x = cos x / sin x, wherever the denominator is not 0.Example: tan(pi/4) = (sqrt(2)/2)/(sqrt(2)/2) = 1
Pythagorean identity: sin^2 x + cos^2 x = 1 for every x. It comes from x^2 + y^2 = 1 on the unit circle.Example: If sin x = 3/5 in quadrant I, cos x = 4/5
Other Pythagorean forms: Divide sin^2 + cos^2 = 1 by cos^2 to get 1 + tan^2 x = sec^2 x, or by sin^2 to get 1 + cot^2 x = csc^2 x.Example: Useful when an integral or limit contains sec^2 x
Double angle for sine: sin(2x) = 2 sin x cos x.Example: sin(2pi/3) = 2 sin(pi/3) cos(pi/3) = 2(sqrt(3)/2)(1/2) = sqrt(3)/2
Double angle for cosine: cos(2x) = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x. Pick the version that matches the rest of the problem.Example: If cos x = 1/3, then cos(2x) = 2/9 - 1 = -7/9
Sum formula for sine: sin(a + b) = sin a cos b + cos a sin b. For a difference, the plus between the products becomes minus.Example: sin(75 degrees) = sin45 cos30 + cos45 sin30 = (sqrt(6) + sqrt(2))/4
Sum formula for cosine: cos(a + b) = cos a cos b - sin a sin b. The sign between the products is opposite to the sign between the angles.Example: cos(a - b) = cos a cos b + sin a sin b
Proving an identity: Work on one side only, rewriting with known identities until it matches the other side. Do not move terms across the equals sign.Example: Show (1 - cos^2 x)/sin x = sin x by replacing 1 - cos^2 x with sin^2 x
Trap: sin(2x) versus 2 sin x: Doubling an angle does not double its sine. The correct expansion is 2 sin x cos x, which equals 2 sin x only when cos x = 1.Example: sin(pi/2) = 1 but 2 sin(pi/4) = sqrt(2)

Equations

Solve 2sin x - 1 = 0 on [0, 2pi): sin x = 1/2, which happens at pi/6 in quadrant I and 5pi/6 in quadrant II.
Solve 2cos^2 x - cos x - 1 = 0 on [0, 2pi): Factor as (2cos x + 1)(cos x - 1) = 0. cos x = -1/2 gives 2pi/3 and 4pi/3; cos x = 1 gives 0.
Solve sin(2x) = sin x on [0, 2pi): Rewrite as 2 sin x cos x - sin x = 0 and factor: sin x(2cos x - 1) = 0. Solutions: 0, pi, pi/3 and 5pi/3.
Trap: dividing by a trig function: Cancelling sin x from both sides throws away every solution where sin x = 0. Move terms to one side and factor instead.Example: In sin(2x) = sin x, dividing loses x = 0 and x = pi
General solution: Every solution, written by adding whole multiples of the period, such as x = pi/6 + 2k pi where k is any integer.Example: tan x = 1 has x = pi/4 + k pi
Equations with 2x or 3x inside: Solve for the inner angle over a stretched interval (for 2x use [0, 4pi)), then divide by the coefficient.Example: sin(2x) = 0 on [0, 2pi): 2x = 0, pi, 2pi, 3pi, so x = 0, pi/2, pi, 3pi/2
Inverse sine range: arcsin returns angles only between -pi/2 and pi/2, so a calculator shows one answer. Use symmetry and the period to find the rest.Example: sin x = 0.6 gives about 0.644 and also pi - 0.644, about 2.498

Graphs

Amplitude: Half the distance from the maximum to the minimum of a sinusoid. For y = a sin(bx) it equals |a|.Example: y = -3cos x swings between -3 and 3
Period of y = sin(bx): 2pi/|b| for sine and cosine, and pi/|b| for tangent.Example: y = cos(4x) repeats every pi/2
Phase shift: The horizontal translation of a sinusoid. In y = sin(b(x - c)) it is c units, so factor out b first.Example: y = sin(2x - pi) = sin(2(x - pi/2)) shifts right pi/2
Midline: The horizontal line y = d halfway between the maximum and minimum of y = a sin(b(x - c)) + d.Example: y = 2sin x + 5 oscillates about y = 5
Graph of y = tan x: Vertical asymptotes at x = pi/2 + k pi, zeros at multiples of pi, period pi, and no amplitude because it is unbounded.Example: tan x approaches positive infinity just left of pi/2