Presentation on theme: "Simplify an expression"— Presentation transcript:
1Simplify an expression EXAMPLE 3Simplify an expressionSimplify the expression cos (x + π).cos (x + π)= cos x cos π – sin x sin πSum formula for cosine= (cos x)(–1) – (sin x)(0)Evaluate.= – cos xSimplify.
2Solve a trigonometric equation EXAMPLE 4Solve a trigonometric equationSolve sin ( x + ) + sin ( x – ) = 1 for 0 ≤ x < 2π.π3Write equation.sin ( x + ) + sin ( x – )π3= 1Use formulas.sin x cos cos x sin sin x cos – cos x sinπ3= 1sin x cos x sin x – cos x123= 1Evaluate.sin x= 1Simplify.ANSWERIn the interval 0 ≤ x <2π, the only solution is x =π2
3EXAMPLE 5Solve a multi-step problemDaylight HoursThe number h of hours of daylight for Dallas, Texas, and Anchorage, Alaska, can be approximated by the equations below, where t is the time in days and t = 0 represents January 1. On which days of the year will the two cities have the same amount of daylight?Dallas:h1Anchorage:h2π t182= 2 sin ( – 1.35)π t182= –6cos ( )
4EXAMPLE 5Solve a multi-step problemSOLUTIONSTEP 1Solve the equation h1 = h2 for t.2 sin ( – 1.35)π t182π t182= – 6 cos ( )sin ( – 1.35)π t182π t182= – 3 cos ( )sin ( ) cos 1.35 – cos ( ) sin 1.35π t182π t182= – 3 cos ( )sin ( ) (0.219) – cos ( ) (0.976)π t182π t182= – 3 cos ( )0.219 sin ( )π t182π t182= – cos ( )
5EXAMPLE 5Solve a multi-step problemπ t182tan ( )= – 9.242π t182= tan –1 (– 9.242) + nππ t182– nπt– nSTEP 2Find the days within one year (365 days) for which Dallas and Anchorage will have the same amount of daylight.t– (1)97, or on April 8t– (2)279, or on October 7
6GUIDED PRACTICEfor Examples 3, 4, and 5Simplify the expression.6. sin (x + 2π)8. tan (x – π)tan xANSWERsin xANSWER7. cos (x – 2π)cos xANSWER
7GUIDED PRACTICEfor Examples 3, 4, and 5π t75π t759. Solve 6 cos ( ) + 5 = – 24 sin ( ) + 5 for 0 ≤ t < 2π.about 5.65ANSWER