Presentation on theme: "Section 5.3 Trigonometric Functions on the Unit Circle."— Presentation transcript:
Section 5.3 Trigonometric Functions on the Unit Circle
Given a unit circle, the radius or hypotenuse = 1 Sine and Cosine
MEMORIZE THIS: The right angle is always on the x axis. The acute angle is always at the origin. MEMORIZE THIS TOO: The ordered pair of the point where the terminal side of the angle intersects the circle is (x, y) where cosine Ѳ = x and sine Ѳ = y.
45° – 45° – 90° Triangle s s
s s s 45
30° – 60° – 90° Triangle
MEMORIZE THE TRIG RATIOS FOR THE SPECIAL RIGHT TRIANGLES IN THE FIRST QUADRANT These ratios are repeated in each quadrant around the circle, with sign changes.
Signs of the trig functions in the 4 quadrants Quadrant I = All (are positive) Quadrant II = Students (Sine & Cosecant are +) Quadrant I = Take (Tangent & Cotangent are +) Quadrant I = Calculus (Cosine & Secant are +) AS T C
Use the unit circle to find the value for the six trigonometric functions for a 135° angle.
You can apply the Pythagorean theorem to solve for any right triangle.
Consider an angle with a point on its terminating side of (5, -12). That would be in the 4 th quadrant.
Find the values of the six trigonometric functions for an angle Ѳ in standard position if a point with the coordinates (-15,20) lies on its terminal side.
If you know the value of one of the trig. functions and the quadrant in which the terminal side of Ѳ lies, you can find the values of the remaining 5 functions.
Suppose Ѳ is an angle in standard position whose terminal side lies in Quadrant IV. If Find the values of the five remaining functions of Ѳ.
Now try these on page 296 #1-13: 1. Why is csc 180 0 undefined? 2.Show that the value of sin Ѳ increases from 0 0 to 90 0 and decreases from 90 0 to 180 0. 3.Confirm that 4.Complete the chart for the signs of the trig functions in each quadrant. FunctionQuadrant. IQuadrant. IIQuadrant. IIIQuadrant. IV Sin α & Cos α Cos α & Sec α Tan α & Cos α
Use your unit circle to find the exact measure for each of the following. 1.Tan 180 0 2.Sec -90 0 3.Tan 45 0 4.Cot 270 0 5.Tan 135 0 6.Csc 270 0 7.Tan 360 0 8.Sec 180 0
Find the values of the six trig functions for an angle θ in standard position if a point with the given coordinates lies on its terminal side. 1.(3,5) 2.(-6,6)
Use the unit circle to find the sin (-90°)
Use the unit circle to find the cot (270°)
Use the unit circle to find the value for the six trigonometric functions for a 210° angle.