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 Dependent Variable

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# Review of Trigonometric Functions

 Trigonometric Identities [Note that sin2θ is used to represent (sinθ)2.] Pythagorean Identities: Reduction Formulas: Sum or Difference of Two Angles: Half–Angle Formulas: Double–Angle Formulas: Law of Cosines: Reciprocal Identities: Quotient Identities: Evaluating Trigonometric Functions There are two ways to evaluate trigonometric functions: (1) decimal approximations with a calculator (or a table of trigonometric values) and (2) exact evaluations using trigonometric identities and formulas from geometry. When using a calculator to evaluate a trigonometric function, remember to set the calculator to the appropriate mode—degree mode or radian mode.EXAMPLE 2 Exact Evaluation of Trigonometric Functions The angle π/3 in standard position Figure D.32 Evaluate the sine, cosine, and tangent of π/3. Solution Begin by drawing the angle in standard position, as shown in Figure D.32. Then, because radians, you can draw an equilateral triangle with sides of length 1 and θ as one of its angles. Because the altitude of this triangle bisects its base, you know that Using the Pythagorean Theorem, you obtain Now, knowing the values of x, y, and r, you can write the following. NOTE All angles in this text are measured in radians unless stated otherwise. For example, when we write sin 3, we mean the sine of 3 radians, and when we write we mean the sine of 3 degrees. The degree and radian measures of several common angles are given in the table below, along with the corresponding values of the sine, cosine, and tangent (see Figure D.33). Common First Quadrant Angles The quadrant signs for the sine, cosine, and tangent functions are shown in Figure D.34. To extend the use of the table on the preceding page to angles in quadrants other than the first quadrant, you can use the concept of a reference angle (see Figure D.35), with the appropriate quadrant sign. For instance, the reference angle for 3π/4 is π/4 and because the sine is positive in the second quadrant, you can write Similarly, because the reference angle for 330° is  30° and the tangent is negative in the fourth quadrant, you can write  Figure D.35 Common angles Figure D.33 Quadrant signs for trigonometric functions Figure D.34

 EXAMPLE 3 Trigonometric Identities and Calculators Evaluate the trigonometric expression. Solution a. Using the reduction formula you can write b. Using the reciprocal identity you can write c. Using a calculator, you can obtain Remember that 1.2 is given in radian measure. Consequently, your calculator must be set in radian mode. Solving Trigonometric Equations How would you solve the equation ? You know that is one solution, but this is not the only solution. Any one of the following values of θ is also a solution. You can write this infinite solution set as is { nπ :n an integer }

 EXAMPLE 4 Solving a Trigonometric Equation Solution points of Figure D.36 Solve the equation Solution To solve the equation, you should consider that the sine is negative in Quadrants III and IV and that Thus, you are seeking values of θ in the third and fourth quadrants that have a reference angle of π/3In the interval ,the two angles fitting these criteria are and By adding integer multiples of 2π to each of these solutions, you obtain the following general solution. ,where n is an integer. (See Figure D.36.)