Trigonometry Chapter Review

Co-terminal Angles

• Co-terminal angles can be determined by adding or subtracting multiples of ±360º or ±2π.

Radians in a Full Revolution

• One full revolution equals 6.28 radians, or equivalently, 2π radians.

Conversion Formulas

• To convert degrees to radians, multiply degrees by π / 180º.
• To convert radians to degrees, multiply radians by 180º / π.

Arc Length Formulas

• When measuring in degrees, arc length is calculated as (θº/360º) ∙ 2πr, where θ is in degrees and r is the radius.
• In radians, arc length is found using the formula arc length = rθ, where θ is in radians.

Reference Angle

• The reference angle is the acute angle created between the terminal side of an angle and the x-axis; it is always positive.

Quadrants and Trigonometric Ratios

• In Quadrant I, all trigonometric ratios are positive.
• In Quadrant II, only the sine function is positive.
• In Quadrant III, only the tangent function is positive, as sine and cosine are both negative.
• In Quadrant IV, only the cosine function is positive.

Key Trigonometric Functions

• Secant: ( \sec \theta = \frac{1}{\cos \theta} )
• Cosecant: ( \csc \theta = \frac{1}{\sin \theta} )
• Cotangent: ( \cot \theta = \frac{1}{\tan \theta} )
• Tangent: ( \tan \theta = \frac{\sin \theta}{\cos \theta} )
• Cotangent: ( \cot \theta = \frac{\cos \theta}{\sin \theta} )

Pythagorean Identity

• The fundamental Pythagorean identity states that ( \sin^2 \theta + \cos^2 \theta = 1 ).

Wave Properties

• Amplitude of a sine function is represented as |a| in the equation ( y = a \sin(bx + c) + d ).
• General equation for sine graphs is ( y = a \sin(bx + c) + d ).

Period and Shifts

• The period of a sine or cosine function is determined by ( \text{Period} = \frac{2π}{b} ).
• Gap, or interval between key points, is defined as ( \text{period} / 4 ).
• Vertical shift is represented by the value d in the function.
• Phase shift is calculated using ( \text{phase shift} = \frac{c}{b} ).

Graphing Sine Functions

• Starting point of the graph is positioned at the phase shift ( c/b ).
• The endpoint can be determined by adding the period to the starting point.
• The top of the wave is at the vertical shift plus amplitude, while the bottom is at the vertical shift minus amplitude.

Tangent and Cotangent Functions

• The period of tangent and cotangent functions is given by ( \frac{π}{b} ).
• Asymptotes for the tangent function occur at ( (±\text{period}/2) ± \text{period} ).
• Asymptotes for the cotangent function can be found at ( 0 ± π ).