Trigonometric Functions

Cosecant is reciprocal of sine. Secant is reciprocal of cosine. Cotangent is reciprocal of tangent.

Suppose the terminal side of a rotation angle (K) passes through the point (–3, 4). Draw angle K in standard position. Find the distance from (–3, 4) to the origin using Evaluate the six trig functions for this angle.

Suppose angle K measures approximately ―233.13°. Do the calculator values for sine, cosine, and tangent of this angle match the answers you previously found for angle K?

A reference triangle can be formed by drawing a perpendicular segment from a point on the angle’s terminal side to the x-axis. This perpendicular segment (a y-value) can act as the “opposite” side, and its distance from the distance from origin (an x-value) serves as the “adjacent” side. Here, however, the opposite and adjacent legs can be negative. x = “adjacent” leg y = “opposite” leg r = “hypotenuse” These values, because they are coordinates, can be positive or negative (or even zero). The r-value, however, because it is a distance, is defined to always be positive.

Angle: θ, Point: (12, –5) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using C) Evaluate the six trig functions for this angle. Quadrant IV

Angle: α, Point: (–1, 2) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using C) Evaluate the six trig functions for this angle. Quadrant II

Angle: Z, Point: (–3, –3) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using C) Evaluate the six trig functions for this angle. Quadrant III

Angle: P, Point: (24, 7) A) Sketch the angle in standard position, and indicate the Quadrant where the terminal side lies. B) Find the value of r using C) Evaluate the six trig functions for this angle. Quadrant I

Based on the answers to these problems, tell in which Quadrant the following are always true. Only sine (and cosecant) values are positive Only cosine (and secant) values are positive Only tangent (and cotangent) values are positive All trig ratio values are positive Quadrant I Quadrant III Quadrant IV Quadrant II Label the Quadrants in which the six trig functions are positive.

A) Sketch the angle in standard position B) Find the missing value C) Evaluate the other trig functions for this angle x’s are negative in Quad II