7.3 Sum and Difference Identities 7-1 Cosine Sum and Difference Identities: cos A B Cosine of a Sum or Difference cos cos does NOT equal cos A cos B. AB AB EXAMPLE 1 Finding Eact Cosine Function Values Find the eact value of each epression. (a) cos15 (b) cos 75 Sine of a Sum or Difference sin sin Tangent of a Sum or Difference AB AB tan A B tan AB EXAMPLE 3 Finding Eact Sine and Tangent Function Values Find the eact value of each epression. (a) sin 75 (b) tan 105 (c) sin 40 cos 160 cos 40 sin 160 (d) cos87 cos93 sin87 sin93
EXAMPLE 4 Writing Functions as Epressions Involving Functions of Write each function as an epression involving functions of. cos 30 tan 45 (a) (b) (c) sin180 EXAMPLE 5 Finding Function Values and the Quadrant of A + B 4 Suppose that A and B are angles in standard position, with sin A, A, and 5 Find each of the following. (a) sin (A + B) (b) tan (A + B) 5 3 cos B, B. 13
Verifing an Identit 7-3 EXAMPLE 7 Verifing an Identit Verif that the following equation is an identit. sin cos cos 6 3
7.4 Double-Angle and Half-Angle Identities Double-Angle Identities Verifing an Identit Double-Angle Identities cos A = sin A = cos A = cos A = tan A = EXAMPLE 1 Finding Function Values of Given Information about 3 Given cos and sin 0, find sin, cos, and tan. 5 EXAMPLE Finding Function Values of Given Information about 4 Find the values of the si trigonometric functions of if cos and 90 180. 5
EXAMPLE 3 Simplifing Epressions Using Double-Angle Identities Simplif each epression. (a) cos 7 sin 7 (b) sin15cos15 7-5 EXAMPLE 4 Deriving a Multiple-Angle Identit Write sin 3 in terms of sin. Half-Angle Identities In the following identities, the smbol indicates that the sign is chosen based on the function A under consideration and the of. cos ( A ) = sin (A ) = tan (A ) = EXAMPLE 9 Using a Half-Angle Identit to Find an Eact Value A sin A Find the eact value of tan.5 using the identit tan. 1 cos A
7.5 Inverse Circular Functions Review of Inverse Functions Inverse Sine, Cosine, Tangent Functions Remaining Inverse Circular Functions Inverse Function Values Review of Inverse Functions If a function is defined so that each element is used, then it is called a one-to-one function. Do not confuse the 1 in represents the of f. 1 f 1 with a negative eponent. The smbol f does NOT represent 1. f It Review of Inverse Functions 1. In a one-to-one function, each -value corresponds to -value and each -value corresponds to -value.. If a function f is one-to-one, then f has an f 1. 3. The domain of f is the of f 1, and the range of f is the of point (a, b) is on the graph of f, then is on the graph of 4. The graphs of f and 1 5. To find f Step 1 Replace Step Solve for. Step 3 Replace with f 1. 1 f are of each other across the line =. from f, follow these steps. f with and interchange and. f 1. f 1. That is, if the Inverse Sine Function 1 sin or arcsin means that sin, for. 1 We can think of sin or arcsin as is the angle in the interval, whose sine is. EXAMPLE 1 Finding Inverse Sine Values Find in each equation. (a) 1 arcsin 1 1 (b) sin 1 (c) sin Be certain that the number given for an inverse function value is in the range of the particular inverse function being considered.
7-7 Domain: Range: Inverse Cosine Function 1 cos or arccos means that cos, for 0. Domain: Range: EXAMPLE Finding Inverse Cosine Values Find in each equation. (a) = arccos (-1) (b) cos 1 (c) = arccos (1/)
Inverse Tangent Function 1 tan or arctan means that tan, for. Domain: Range: Remaining Inverse Circular Functions Inverse Cotangent, Secant, and Cosecant Functions 1 cot or arccot 1 sec or arcsec 1 csc or arccsc means that cot, for 0. means that sec, for 0,. means that csc, for, 0.
Inverse Function Domain Interval 7-9 Range Quadrants of the Unit Circle sin 1 cos 1 tan 1 cot 1 sec 1 csc 1 EXAMPLE 3 Finding Inverse Function Values (Degree-Measured Angles) Find the degree measure of in the following. (a) arctan1 (b) 1 sec Use the following to evaluate these inverse trigonometric functions on a calculator. sec 1 can be evaluated as cot 1 cos ; can be evaluated as csc 1 1 1 can be evaluated as 1 1 1 tan if 0 1 180 tan if 0. sin ; 1 1 EXAMPLE 4 Finding Inverse Function Values with a Calculator Use a calculator to give each value. 1 (a) Find in radians if csc 3. (b) Find in degrees if arccot 0.3541. Be careful when using our calculator to evaluate the inverse cotangent of a negative quantit. To do this, we must enter the inverse tangent of the of the negative quantit, which returns an angle in quadrant. Since inverse cotangent is in quadrant II, adjust our calculator 1 result b adding 180 or accordingl. Note that cot 0.
EXAMPLE 5 Finding Function Values Using Definitions of the Trigonometric Functions Evaluate each epression without using a calculator. (a) sin (sin 1 (0.7)) (b) sin 1 (sin ( 3π 4 )) (c) sintan 1 3 (d) tancos 13 1 5 EXAMPLE 6 Finding Function Values Using Identities Evaluate each epression without using a calculator. (a) 1 cosarctan 3 arcsin 3 (b) tan arcsin 5
7-11 7.6 Solving Trigonometric Equations Solving b Linear Methods Solving b Factoring Solving b Quadratic Methods Solving b Using Trigonometric Identities Equations with Half-Angles Equations with Multiple Angles Applications Solving b Linear Methods EXAMPLE 1 Solving a Trigonometric Equation b Linear Methods Solve the equation sin 1 0 (a) over the interval [0, 360 ), and (b) for all solutions. Solving b Factoring EXAMPLE Solving a Trigonometric Equation b Factoring Solve sin tan sin over the interval [0, 360 ).
Solving b Quadratic Methods EXAMPLE 3 Solving a Trigonometric Equation b Factoring Solve tan tan 0 0,. over the interval EXAMPLE 4 Solving a Trigonometric Equation Using the Quadratic Formula cot cot 3 1. Write the solution set. Find all solutions of
Solving a Trigonometric Equation 1. Decide whether the equation is or in form, so that ou can determine the solution method.. If onl one trigonometric function is present, for that function. 7-13 3. If more than one trigonometric function is present, rearrange the equation so that one side equals. Then tr to and set each equal to 0 to solve. 4. If the equation is quadratic in form, but not factorable, use the. Check that solutions are in the desired interval. 5. Tr using to change the form of the equation. It ma be helpful to square each side of the equation first. In this case, check for solutions. Equations with Half-Angles EXAMPLE 6 Solving an Equation with a Half-Angle Solve the equation sin 1 (a) over the interval 0,, and (b) for all solutions. Equations with Multiple Angles EXAMPLE 7 Solving an Equation Using a Double Angle Solve cos() = over the interval 0,.
EXAMPLE 8 Solving an Equation Using a Multiple-Angle Solve sin(3) = 1 (a) over the interval [0, 360 ), and (b) for all solutions.