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Algebra 2 Unit 8: Trigonometric functions

Algebra 2 extends trig from right triangles to any angle using the unit circle, and introduces sine and cosine as periodic functions.

Covers: Radians, unit circle, sine and cosine values, graphs of sine and cosine, coterminal and reference angles

Study guide

Radians measure angles by arc length on a circle of radius 1: one full turn is 2π radians, so 180° = π radians. To convert degrees to radians multiply by π/180, and to convert radians to degrees multiply by 180/π.

The unit circle is a circle of radius 1 centered at the origin. For an angle θ in standard position (starting on the positive x-axis, counterclockwise for positive angles), the point where it meets the circle is (cos θ, sin θ). So cosine is the x-coordinate and sine is the y-coordinate, and tan θ = sin θ/cos θ. Memorize the values at 0, π/6, π/4, π/3 and π/2 (0°, 30°, 45°, 60°, 90°), then use symmetry for the other quadrants: all are positive in Quadrant I, only sine in II, only tangent in III, and only cosine in IV.

Coterminal angles share a terminal side and differ by multiples of 360° (2π). A reference angle is the acute angle between the terminal side and the x-axis; the trig values of any angle equal those of its reference angle, with the sign set by the quadrant.

The graphs of y = sin x and y = cos x are waves with amplitude 1 and period 2π. In y = a sin(b(x − h)) + k, |a| is the amplitude, 2π/|b| is the period, h is the phase shift, and k is the midline.

On the unit circle, sin²θ + cos²θ = 1. If you know one value, solve for the other, choose its sign from the quadrant, and find tan θ = sin θ/cos θ. Frequency is the number of cycles per unit of input and equals 1/period. In y = a sin(bx), the period is 2π/|b|, so a larger |b| means a higher frequency and a shorter period.

To find when a sinusoidal model reaches a given value, set the function equal to that value and solve by graphing both sides and finding the intersections in the required interval. There are usually two solutions per cycle. cos(−θ) = cos θ, so cosine is even. sin(−θ) = −sin θ and tan(−θ) = −tan θ, so sine and tangent are odd.

Key ideas

• Radians measure angles by arc length: 180° = π.
• On the unit circle, (cos θ, sin θ) is the point at angle θ.
• Coterminal angles differ by multiples of 360° (or 2π).
• Reference angles and quadrant signs give values for any angle.
• y = sin x and y = cos x have period 2π and amplitude 1.

Formulas and rules

Common mistakes

• Calculator in the wrong mode.
• Wrong signs outside quadrant I.

Words to know

Degrees to radians

Multiply by π/180 (90° = π/2).

sin 90°

1

cos 180°

−1

Coterminal angles

Angles that differ by a multiple of 360° (2π).

Period of y = cos x

2π

Unit circle

The point at angle θ is (cos θ, sin θ).

Radian

An angle measure: 180° = π radians.

Standard position

An angle starting on the positive x-axis, opening counterclockwise.

Reference angle

The acute angle between the terminal side and the x-axis.

Quadrant signs

All positive in I, sine in II, tangent in III, cosine in IV.

Amplitude

|a| in y = a sin x: half the max-to-min distance.

Period of sin(bx)

2π/|b|

Midline

y = k in y = a sin(bx) + k.

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.