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Trigonometric Functions: Definitions, Values, and Identities

Learn about the six trigonometric functions, their values for different angles, and important identities such as reciprocal and quotient identities.

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Trigonometric Functions: Definitions, Values, and Identities

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  1. Chapter 8: The Unit Circle and the Functions of Trigonometry 8.1 Angles, Arcs, and Their Measures 8.2 The Unit Circle and Its Functions 8.3 Graphs of the Sine and Cosine Functions 8.4 Graphs of the Other Circular Functions 8.5 Functions of Angles and Fundamental Identities 8.6 Evaluating Trigonometric Functions 8.7 Applications of Right Triangles 8.8 Harmonic Motion

  2. 8.5 Functions of Angles and Fundamental Identities • To define the six trigonometric functions, start with an angle  in standard position. Choose any point P having coordinates (x,y) on the terminal side as seen in the figure below. • Notice that r > 0 since distance is never negative.

  3. 8.5 Trigonometric Functions • The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Trigonometric Functions Let (x,y) be a point other than the origin on the terminal side of an angle  in standard position. The distance from the point to the origin is The six trigonometric functions of angle  are as follows.

  4. 8.5 Finding Function Values of an Angle Example The terminal side of angle  (beta) in standard position goes through (–3,–4). Find the values of the six trigonometric functions of . Solution

  5. 8.5 Finding Function Values of an Angle • The six trigonometric functions can be found from any point on the line. Due to similar triangles, so sin = y/r is the same no matter which point is used to find it.

  6. 8.5 Function Values of Quadrantal Angles Undefined Function Values If the terminal side of a quadrantal angle lies along the • y-axis, the tangent and secant functions are undefined. • x-axis, the cotangent and cosecant functions are undefined.

  7. 8.5 Function Values of Quadrantal Angles

  8. 8.5 Reciprocal Identities • Since sin = y/r and csc = r/y, Similarly, we have the following reciprocalidentities for any angle  that does not lead to a 0 denominator.

  9. 8.5 Using the Reciprocal Identities Example Find sin if csc = Solution

  10. 8.5 Signs and Ranges of Function Values • In the definitions of the trigonometric functions, the distance r is never negative, so r > 0. • Choose a point (x,y) in quadrant I, then both x and y will be positive, so the values of the six trigonometric functions will be positive in quadrant I. • A point (x,y) in quadrant II has x < 0 and y > 0. This makes sine and cosecant positive for quadrant II angles, while the other four functions take on negative values. • Similar results can be obtained for the other quadrants.

  11. 8.5 Signs and Ranges of Function Values Example Identify the quadrant (or quadrants) of any angle  that satisfies sin > 0, tan < 0. Solution Since sin > 0 in quadrants I and II, while tan < 0 in quadrants II and IV, both conditions are met only in quadrant II.

  12. 8.5 Signs and Ranges of Function Values • The figure shows angle  as it increases from 0º to 90º. • The value y increases as  increases, but never exceeds r, so y  r. Dividingboth sides by r gives • In a similar way, angles in quadrant IV suggests

  13. 8.5 Signs and Ranges of Function Values • Since for any angle . • In a similar way, • sec and csc are reciprocals of sin and cos, respectively, making

  14. 8.5 Ranges of Trigonometric Functions • For any angle  for which for which the indicated function • exists: • –1  sin  1 and –1  cos  1; • tan and cot may be equal to any real number; • sec  –1 or sec  1 and csc  –1 or csc  1. Example Decide whether each statement is possible or impossible. • (b) tan = 110.47 (c) sec = 0.6 Solution • Not possible since • Possible since tangent can take on any value. • Not possible since sec  –1 or sec  1.

  15. 8.5 Pythagorean Identities • Three new identities from x2 + y2 = r2 • Divide by r2 Since cos = x/r and sin = y/r, this result becomes • Divide by x2 • Dividing by y2 leads to 1 + cot2 = csc2.

  16. 8.5 Pythagorean Identities Pythagorean Identities

  17. 8.5 Quotient Identities • Recall that Consider the quotient of sin and cos where cos  0. • Similarly Quotient Identities

  18. 8.5 Quotient Identities Example Find sin and cos, if tan = 4/3 and  is in quadrant III. Solution Since  is in quadrant III, sin and cos will both be negative.

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