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Section 5.3

Section 5.3 . Circular Functions. 5.3 Notes We know how to find the values of sine and cosine of an acute angle in a Right Triangle:. 5.3 Notes Using our knowledge of right triangle trig, we want to define cosine and sine in terms of the unit circle.

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Section 5.3

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  1. Section 5.3 Circular Functions

  2. 5.3 Notes • We know how to find the values of sine and cosine of an acute angle in a Right Triangle:

  3. 5.3 Notes • Using our knowledge of right triangle trig, we want to define cosine and sine in terms of the unit circle. • We can draw a vertical line from the point, P(x,y), to the x-axis to form a right triangle with angle, q.

  4. 5.3 Notes • What does the hypotenuse of the right triangle equal? • The hypotenuse = 1 because the radius of the unit circle is 1. • In this diagram, what is cos(q) and sin(q)? If the terminal side of an angle, q, in standard position intersects the unit circle at a point, P(x,y), then cos(q)=x and sin(q)=y.

  5. 5.3 Notes • What do you predict tan(q) is equal to?

  6. 5.3 Notes • When a point P(x,y) lies on the intersection of the terminal side of some angle, q, and the unit circle; the six trig functions are as follows:

  7. 5.3 Notes • If you are given a point, P’(x’,y’), on the terminal side of an angle, q, and NOT on the unit circle, we need to set up a proportion to solve for the six trig functions.

  8. 5.3 Notes • Using properties of similar triangles, we can set up a proportion to solve for cos(q) and sin(q). • r is the hypotenuse of the triangle with legs x’ and y’. • Using the pythagorean theorem:

  9. 5.3 Notes • Given a point P(x,y) on the terminal side of some angle, q, and NOT on the unit circle, the six trig functions are found as follows:

  10. 5.3 Notes • What are the six trig function values given the point (3,4) which lies on the terminal side of some angle, q? • First we need to solve for r. • Now we can solve for the six trig functions.

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