Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
14.3 Verify Trigonometric Identities Goal Verify trigonometric identities. Your Notes VOCABULARY Trigonometric identity A trigonometric equation that is true for all values of (in its domains) FUNDAMENTAL TRIGONOMETRIC IDENTITIES Reciprocal Identities csc = 1 sin sec = 1 cos cot = 1 tan Tangent and Cotangent Identities tan = sin cos cot = cos sin Pythagorean Identities sin2 + cos2 = __1__ 1+ tan2 =__sec2 __ 1+cot2 =__csc2__ Cofunction Identities Negative Angle Identities sin =__cos __ 2 sin () =__sin__ cos =__sin __ 2 cos () = __cos__ tan =__cot __ 2 tan () = __tan __ Your Notes Example 1 Find trigonometric values 3 Given that cos = 3 and < < , find the values of the other five trigonometric 2 4 functions of . Solution Step 1 Find sin . sin 2 +cos 2 =1 Write Pythagorean Identity. 2 3 sin2 + =1 4 3 for cos . 4 2 3 Subtract from each side. 4 Substitute sin2 = 1 3 4 2 7 sin2 = _____ 16 7 sin = 4 Simplify. Take square roots of each side. Because is in Quadrant __III__, sin is negative. So, sin = 7 4. Step 2 Find the values of the other four trigonometric functions of using the known values of sin and cos . 7 tan = sin 4 7 3 cos 3 4 3 cot = cos 4 3 3 7 sin 7 7 7 csc = 1 14 4 4 7 sin 7 7 7 4 sec = 1 1 4 cos 3 3 4 Your Notes Checkpoint Find the values of the other five trigonometric functions of . 1 ,0 4 2 sin = 15 ; tan = 15 ; cot = 15 ; 4 15 csc = 4 15 ; sec = 4 1. cos = 15 1 3 2. sin = , 2 3 2 cos = 2 2 ; tan = 2 2 ; cot = 2 2 ; 3 8 csc = 3; sec = 6 2 8 Example 2 Simplify a trigonometric expression 1 cot . sin 0 2 Solution 1 cot 1 = cot . cos ______ sin 0 2 = 1 co ssin cos ______ _____ Simplify the expression = 1 sin _____ = __csc __ Checkpoint Simplify the expression. 3. tan 3 sin + tan csc cos sec 1 ___Cofunction identity__ Cotangent identity Simplify. __Reciprocal identity__ Your Notes Example 3 Verify a trigonometric identity Verify the identity 1 cos 2 1 sec tan cot = . 1 sin 2 cos b Solution sec 1 tan cot = sec __sec __ tan cot cos = __sec2 _ tan cot 1 = ___sec2 __ tan tan _____ = __sec2 _ __1__ = __tan2 __ sin2 = _______ cos2 2 = 1 cos 2 1 sin Checkpoint Verify the identity. 4. cos 2 ( x ) sin 2 x 2 cot x cos 2 ( x) cos2 x cot 2 x cot 2 x cos 2 x = cos 2 x sin 2 x sin 2 x cos 2 x cos 2 x = sin2 x Homework ________________________________________________________________________ ________________________________________________________________________