Test 2 Trigonometric Study Guide

Period of sin(t) and cos(t)

2π (Full revolution is 2π)

Period of tan(t) and cot(t)

π (Full revolution is π)

Period of sec(t) and csc(t)

2π (Full revolution is 2π)

sin(t + even·π)

sin(t) (Even multiples of π return to the exact same value)

sin(t + odd·π)

-sin(t) (Odd multiples of π reflect across the origin)

cos(t + even·π)

cos(t) (Even multiples of π return to the exact same value)

cos(t + odd·π)

-cos(t) (Odd multiples of π negate the cosine value)

tan(t + any·π)

tan(t) (Adding any integer multiple of π leaves tangent unchanged)

sin(

t) - -sin(t) (Sine is an ODD function)

cos(

t) - cos(t) (Cosine is an EVEN function)

tan(

t) - -tan(t) (Tangent is an ODD function)

Reciprocal product: sin(t) · csc(t)

1 (Because csc(t) = 1/sin(t))

Reciprocal product: cos(t) · sec(t)

1 (Because sec(t) = 1/cos(t))

Reciprocal product: tan(t) · cot(t)

1 (Because cot(t) = 1/tan(t))

Primary Pythagorean Identity

sin²(t) + cos²(t) = 1

Pythagorean Identity for tangent and secant

tan²(t) + 1 = sec²(t)

Pythagorean Identity for cotangent and cosecant

1 + cot²(t) = csc²(t)

Alternate forms of primary Pythagorean identity

1 - sin²(t) = cos²(t) and 1 - cos²(t) = sin²(t)

General Wave Equation Form

f(x) = A·sin(Bx - C) + D or f(x) = A·cos(Bx - C) + D

What does "A" represent?

Amplitude / vertical stretch: Amplitude = |A| = (Max - Min) / 2

What does "B" represent?

Frequency factor: B = 2π / Period (for sin, cos) or π / Period (for tan)

What does "C" represent?

Horizontal shift parameter: Phase Shift = C / B (shifts right if C/B > 0, left if C/B < 0)

What does "D" represent?

Midline / vertical shift: Midline is y = D, calculated as D = (Max + Min) / 2

Period formula for sin(x) and cos(x)

Period = 2π / B

Period formula for tan(x) and cot(x)

Period = π / B

Maximum value formula

Max = D + |A|

Minimum value formula

Min = D - |A|

Where parent sin(x) starts on y

axis (at x = 0) - At the midline: (0, 0), heading upwards

Where parent cos(x) starts on y

axis (at x = 0) - At the peak / maximum: (0, 1), heading downwards

Where parent tan(x) starts on y

axis (at x = 0) - At the origin: (0, 0), heading upwards between asymptotes at ±π/2

Transforming point coordinates (x, y)

x new = (x old + C) / B (or solve Bx - C = x old); y new = A · y old + D

Vertical asymptotes of tan(x) and sec(x)

Set inside argument Bx - C = π/2 + kπ (where denominator cos = 0)

Vertical asymptotes of cot(x) and csc(x)

Set inside argument Bx - C = kπ (where denominator sin = 0)

Restricted range / output interval of arcsin(x)

[-π/2, π/2] (Quadrants I and IV)

Restricted range / output interval of arccos(x)

[0, π] (Quadrants I and II)

Restricted range / output interval of arctan(x)

(-π/2, π/2) (Quadrants I and IV)

Evaluating composite trig like tan(arccos(2/5))

Draw a right triangle: adjacent = 2, hypotenuse = 5, opposite = √(25 - 4) = √21 -> tan = √21 / 2