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Section 5.3 Trigonometric Functions on the Unit Circle

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Standard position

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Unit Circle

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Given a unit circle, the radius or hypotenuse = 1 Sine and Cosine

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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.

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45° – 45° – 90° Triangle s s

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s s s 45

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30° – 60° – 90° Triangle

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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.

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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

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Use the unit circle to find the value for the six trigonometric functions for a 135° angle.

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You can apply the Pythagorean theorem to solve for any right triangle.

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Consider an angle with a point on its terminating side of (5, -12). That would be in the 4 th quadrant.

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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.

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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.

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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.

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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

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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

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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)

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Use the unit circle to find the sin (-90°)

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Use the unit circle to find the cot (270°)

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Use the unit circle to find the value for the six trigonometric functions for a 210° angle.

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