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Trigonometric Functions

Trigonometric Functions. Of Any Angle. Unit Circle Rationale. Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y . What we didn’t point out is that since the radius (hypotenuse) is 1, the trig

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Trigonometric Functions

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  1. Trigonometric Functions Of Any Angle

  2. Unit Circle Rationale Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cos θ = and sin θ = . S So what if the radius (hypotenuse) is not 1?

  3. Trig Function of Any Angle Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ. We must first find the hypotenuse (r) by using the Pythagorean Theorem.

  4. Trig Function of Any Angle The six trig functions of a unit circle, where , are:

  5. Trig Function of Any Angle The six trig functions of any angle, where

  6. The Signs of the Trig Functions Always look for the quadrant where the terminal side of the angle is located. Quadrant I Quadrant II Positive sin, csc Positive ALL ALL STUDENTS TAKE CALCULUS Positive cos, sec Positive tan, cot Quadrant III Quadrant IV

  7. P Notice that is an obtuse angle. In which quadrant does the terminal side lie? Check for the validity of the signs in the answers. Let be a point on the terminal side of . Find each of the six trigonometric functions of . Note: We must first find by using the Pythagorean Theorem.

  8. Step-by-step, inch-by-inch • What is the coordinate? Identify x and y. • Find r by using the equation • Substitute the values for x, y, and r into the six trig identities of any triangle. • Check the signs of your answers by using, “ALL STUDENTS TAKE CALCULUS”. (The terminal side of the angle determines which quadrant to use.)

  9. In which quadrant does the terminal side lie? • Check for the validity of the signs in the answers. Let be a point on the terminal side of . Find each of the six trigonometric functions of . Note: We must first find by using the Pythagorean Theorem.

  10. In which quadrant does the terminal side lie? • In which quadrant does the terminal side lie? • Check for the validity of the signs in the answers. Let be a point on the terminal side of . Find each of the six trigonometric functions of . Note: We must first find by using the Pythagorean Theorem.

  11. Trig Functions of Quadrantal Angles Step-by-Step Inch-by-Inch Evaluate the sine function at the following four quadrantal angles. First, we must find r.

  12. Trig Functions of Quadrantal Angles Evaluate the tangent function at the following four quadrantal angles. Since we only need x and y to find tangent, we don’t need to find r. Would the answers change if the angles were expressed as degrees? No

  13. The Signs of the Trig Functions If and , name the quadrant in which angle lies. Quadrant I Quadrant II In which quadrants is tangent negative? Positive sin, csc Positive ALL ALL STUDENTS TAKE CALCULUS In which quadrants is cosine positive? Positive cos, sec Positive tan, cot Quadrant III Quadrant IV Angle lies in quadrant IV

  14. The Signs of the Trig Functions If and , name the quadrant in which angle lies. Quadrant I Quadrant II • In which quadrants is sine negative? Positive sin, csc Positive ALL ALL STUDENTS TAKE CALCULUS • In which quadrants is cosine negative? Positive cos, sec Positive tan, cot Quadrant III Quadrant IV • Angle lies in quadrant III

  15. Evaluating Trigonometric Functions Given and , find and . Step 1: Find the quadrant where the angle lies. Tangent is negative in quadrants II and IV. Cosine is positive in quadrants I and IV. The angle lies in quadrant IV, where the coordinate is Step 2: Find x and y. Step 3: Find r.

  16. Evaluating Trigonometric Functions (Cont’d.) Given and , find and . Step 3: We now know that and . Check: cosine is positive in quadrant IV. Check: Cosecant is negative in quadrant IV.

  17. Evaluating Trigonometric Functions Given and , find and . • Step 1: Find the quadrant where the angle lies. Tangent is negative in quadrants II and IV. Cosine is negative in quadrants II and III. The quadrant is II, therefore, the x and y-coordinate for tangent is . • Step 2: Find x and y. • Step 3: Find r.

  18. Evaluating Trigonometric Functions (Cont’d.) Given and , find and . • Step 3: We now know that and . Check: Sine is positive in quadrant II. • Check: Cosecant is negative in quadrant II.

  19. Reference Angles The reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis.

  20. Reference Angles

  21. Reference Angles

  22. Find the reference angle , for

  23. Finding Reference Angles for Angles Greater Than 360° or Less Than • Find a positive angle less than 360° or that is coterminal with the given angle. • Draw angle in standard position. • Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of and the x-axis is the reference angle.

  24. Find the reference angle , for

  25. Evaluating Trig Functions Using Reference Angles Use reference angles to find the exact value of . Step 1: Find the reference angle of , which is Step 2: Find the quadrant that lies in, which is quadrant II. Step 3: The sine function is positive in quadrant II. Therefore,

  26. Evaluating Trig Functions Using Reference Angles Use reference angles to find the exact value of . Step 1: Find the reference angle of , which is Step 2: Find the quadrant that lies in, which is quadrant III. Step 3: The cosine function is negative in quadrant III. Therefore,

  27. Evaluating Trig Functions Using Reference Angles Use reference angles to find the exact value of . Step 1: Find the reference angle of , which is Step 2: Find the quadrant that lies in, which is quadrant IV. Step 3: The cotangent function is negative in quadrant IV. Therefore,

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