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Published byPhilip Gatling Modified over 3 years ago

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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**

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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45 s s 45 s

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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 +) S A 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)**

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

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:**

Why is csc 1800 undefined? Show that the value of sin Ѳ increases from 00 to 900 and decreases from 900 to Confirm that Complete the chart for the signs of the trig functions in each quadrant. Function Quadrant. I Quadrant. II Quadrant. III Quadrant. 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.**

Tan 1800 Sec -900 Tan 450 Cot 2700 Tan 1350 Csc 2700 Tan 3600 Sec 1800

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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. (3,5) (-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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4.2, 4.4 – The Unit Circle, Trig Functions The unit circle is defined by the equation x 2 + y 2 = 1. It has its center at the origin and radius 1. (0,

4.2, 4.4 – The Unit Circle, Trig Functions The unit circle is defined by the equation x 2 + y 2 = 1. It has its center at the origin and radius 1. (0,

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