Lesson 13.4, For use with pages 875-880 1.cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 5π 6...

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Lesson 13.4, For use with pages 875-880 1. cos 45º ANSWER 1 2 Evaluate the expression. 2. sin 6 3. tan(– 60º) ANSWER 3 ANSWER 2 2

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Lesson 13.4, For use with pages 875-8801.cos 45ANSWER12Evaluate the expression.2. sin 563.tan( 60)ANSWER 3ANSWER221ANSWER1Lesson 13.4, For use with pages 875-880ANSWER33 5. tan 6Evaluate the expression.4.cos 2Trigonometry, Inverse Functions

EXAMPLE 1Evaluate inverse trigonometric functionsEvaluate the expression in both radians and degrees.a.cos132SOLUTIONa.When 0 or 0 180, the angle whose cosine is 32cos132 =6=cos132 ==30

EXAMPLE 1Evaluate inverse trigonometric functionsEvaluate the expression in both radians and degrees.b.sin12SOLUTIONsin1b.There is no angle whose sine is 2. So, is undefined.2

EXAMPLE 1Evaluate inverse trigonometric functionsEvaluate the expression in both radians and degrees.3 ( c.tan1SOLUTIONc.When < < , or 90 < < 90, the angle whose tangent is - is:223 ( )tan13=3= ( )tan13 =60 =

EXAMPLE 2Solve a trigonometric equationSolve the equation sin = where 180 < < 270.58SOLUTIONSTEP 1

sine is is sin1 38.7. This5858

Use a calculator to determine that in theinterval 90 90, the angle whoseangle is in Quadrant IV, as shown.

EXAMPLE 2Solve a trigonometric equationSTEP 2Find the angle in Quadrant III (where180 < < 270) that has the same sinevalue as the angle in Step 1. The angle is:

180 + 38.7 = 218.7CHECK :Use a calculator to check the answer. 58sin 218.7

0.625=

GUIDED PRACTICEfor Examples 1 and 2Evaluate the expression in both radians and degrees.1.sin122ANSWER4, 452.cos112ANSWER3, 603.tan1 (1)ANSWER4, 45

GUIDED PRACTICEfor Examples 1 and 2Evaluate the expression in both radians and degrees.4.sin1 ( )126, 30ANSWER

GUIDED PRACTICEfor Examples 1 and 2Solve the equation for270 < < 3605. cos = 0.4; ANSWERabout 293.6180 < < 2706. tan = 2.1; ANSWERabout 244.5270 < < 3607. sin = 0.23; ANSWERabout 346.7

GUIDED PRACTICEfor Examples 1 and 2Solve the equation for180 < < 2708. tan = 4.7; ANSWERabout 258.090 < < 1809. sin = 0.62; ANSWERabout 141.7180 < < 27010. cos = 0.39; ANSWERabout 247.0

EXAMPLE 3Standardized Test Practice

SOLUTIONIn the right triangle, you are given the lengths of the side adjacent to and the hypotenuse, so use the inverse cosine function to solve for .cos =adjhyp= 611

cos 1= 611

56.9The correct answer is C.

ANSWER

EXAMPLE 4Write and solve a trigonometric equationMonster Trucks

A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle of the ramp?

EXAMPLE 4Write and solve a trigonometric equationSOLUTIONSTEP 1

Draw: a triangle that represents the ramp.STEP 2Write: a trigonometric equation that involves the ratio of the ramps height and horizontal length.tan =oppadj= 820

EXAMPLE 4Write and solve a trigonometric equationSTEP 3Use: a calculator to find the measure of .tan1= 820

21.8The angle of the ramp is about 22.ANSWER

GUIDED PRACTICEfor Examples 3 and 4Find the measure of the angle .

11.SOLUTIONIn the right triangle, you are given the lengths of the side adjacent to and the hypotenuse. So, use the inverse cosine function to solve for .cos =adjhyp=49=

63.6

cos149

GUIDED PRACTICEfor Examples 3 and 4Find the measure of the angle .SOLUTIONIn the right triangle, you are given the lengths of the side opposite to and the side adjacent. So, use the inverse tan function to solve for . 12.

tan =oppadj=108

51.3

=tan1108

GUIDED PRACTICEfor Examples 3 and 4Find the measure of the angle .SOLUTIONIn the right triangle, you are given the lengths of the side opposite to and the hypotenuse. So, use the inverse sin function to solve for . 13.

sin =opphyp=512

24.6=sin1512

GUIDED PRACTICEfor Examples 3 and 414.WHAT IF? In Example 4, suppose a monster truck drives 26 feet on a ramp before jumping onto a row of cars. If the ramp is 10 feet high, what is the angle of the ramp?SOLUTIONSTEP 1Draw: a triangle that represents the ramp.STEP 2Write: a trigonometric equation that involves the ratio of the ramps height and horizontal length.tan =oppadj=1026

GUIDED PRACTICEfor Examples 3 and 4STEP 3Use: a calculator to find the measure of .

22.6tan1=1026The angle of the ramp is about 22.6.ANSWER