1.4 - Trigonometric Functions of Any Angle

Objectives Evaluate trigonometric functions of any angle. Use reference angles to evaluate trig functions. Use trig identities to evaluate trig functions Use trig models to solve real world problems

Definitions of trigonometric functions Review Definitions of trigonometric functions Syr Cxr Tyx

Examples: Let (-4, -3) be a point on the terminal side of θ. Evaluate the six trigonometric functions of θ. Let (-5, 12) be a point on the terminal side of θ. Evaluate the six trigonometric functions.

Example: Use the given point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ. (4, 5)

Quadrant angles Review x y θ cos θ sin θ π/2 π 3π/2 2π

Quadrantal Angles Quadrantal angles- angles whose terminal side of θ lies on an axis.

Quadrantal Angles (Continue)

Examples: Evaluate the six trigonometric functions of θ = 270˚.

Trigonometric function signs Students All All 6 trig functions are positive in quadrant 1 Sin/csc Take Calculus Tan/cot Cos/sec

Find the quadrant sinθ > 0; cosθ <0 sinθ < 0; cosθ > 0

Reference Angles When wanting to determine the trigonometric ratios for angles greater than 90˚ (or less than 0˚) must use corresponding acute angles. Reference Angles: (corresponding acute angles) an acute angle θʹ formed by the terminal side of θ and the x-axis.

Reference angle Let θ be an angle in standard position. Its reference angle is the acute angle θ‘ formed by the terminal side of θ and the horizontal axis.

Drawing Reference Angles Find the reference angle θ', and sketch θ and θ' in standard position. 1) θ = -145° 2)

Examples: Find the reference angle θʹ for each angle θ.

Evaluating Trigonometric Functions. Step for evaluating a trigonometric function of any θʹ. Find the reference angle, θʹ. Evaluate the trigonometric function for the angle θʹ. Use the quadrant in which θ lies to determine the sign of the trigonometric function. Quadrant II Quadrant I Quadrant III Quadrant IV

Examples: Evaluate…

Examples: Evaluate…

Examples: Using the formula, estimate the horizontal distance traveled by a golf ball hit at an angle of 40˚ with an initial speed of 125 feet per second.

Example A golf club called a wedge is made to lift a ball high in the air. If a wedge has a 65˚ loft, how far does a ball hit with an initial speed of 100 feet per second travel?

Example: Your marching band’s flag corps makes a circular formation. The circle is 20 feet wide in the center of the football field. Our starting position is 140 feet from the nearer goal line. How far from this goal line will you be after you have marched 120˚ counterclockwise around the circle?

Example: A circular clock gear is 2 inches wide. If the tooth at the farthest right edge of the gear starts 10 inches above the base of the clock, how far above the base is the tooth after the gear rotates 240˚ counterclockwise?