121616949-math.80 - Verify the identity 2 csc(2 θ = sec θ...

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66 Chapter 4 Transcendental Functions Similarly, as angle x increases from 0 in the unit circle diagram, the first coordinate of the point A goes from 1 to 0 then to - 1, back to 0 and back to 1, so the graph of y = cos x must look something like this: - 1 1 π/ 2 π 3 π/ 2 2 π - π/ 2 - π - 3 π/ 2 - 2 π ................................. .. . . . . . . .. ................................................................. . . . . . . . . . . .. ............................................................... .. . . . . . . . . .. ................................................................. . . . . . . . . .. ................................ Exercises Some useful trigonometric identities are in the introduction, on page xiv . 1. Find all values of θ such that sin( θ ) = - 1; give your answer in radians. 2. Find all values of θ such that cos(2 θ ) = 1 / 2; give your answer in radians. 3. Use an angle sum identity to compute cos( π/ 12). 4. Use an angle sum identity to compute tan(5 π/ 12). 5. Verify the identity cos 2 ( t ) / (1 - sin( t )) = 1 + sin( t ). 6.
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Unformatted text preview: Verify the identity 2 csc(2 θ ) = sec( θ ) csc( θ ). 7. Verify the identity sin(3 θ )-sin( θ ) = 2 cos(2 θ ) sin( θ ). 8. Sketch y = 2 sin( x ). 9. Sketch y = sin(3 x ). 10. Sketch y = sin(-x ). 11. Find all of the solutions of 2 sin( t )-1-sin 2 ( t ) = 0 in the interval [0 , 2 π ]. ⇒ 4.2 The Derivative of sin x What about the derivative of the sine function? The rules for derivatives that we have are of no help, since sin x is not an algebraic function. We need to return to the definition of the derivative, set up a limit, and try to compute it. Here’s the definition: d dx sin x = lim Δ x → sin( x + Δ x )-sin x Δ x . Using some trigonometric identies, we can make a little progress on the quotient: sin( x + Δ x )-sin x Δ x = sin x cos Δ x + sin Δ x cos x-sin x Δ x = sin x cos Δ x-1 Δ x + cos x sin Δ x Δ x ....
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