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Chapter 1 | Functions and Graphs
Therefore, the solutions are given by the angles θ such that sin θ =0 or cos 2 θ =1/2. The solutions of the first equation are θ =0, ± π , ±2 π ,…. The solutions of the second equation are θ = π /4, ( π /4)±( π /2), ( π /4)± π ,…. After checking for extraneous solutions, the set of solutions to the equation is θ = nπ and θ = π 4 + nπ 2 , n =0, ±1, ±2,….
Find all solutions to the equation cos(2 θ ) = sin θ .
1.20
Example 1.26 Proving a Trigonometric Identity
Prove the trigonometric identity 1+tan 2 θ = sec 2 θ .
Solution We start with the identity
sin 2 θ +cos 2 θ =1.
Dividing both sides of this equation by cos 2 θ , we obtain sin 2 θ cos 2 θ +1= 1 cos 2 θ . Since sin θ /cos θ = tan θ and 1/cos θ = sec θ , we conclude that tan 2 θ +1= sec 2 θ .
Prove the trigonometric identity 1+cot 2 θ =csc 2 θ .
1.21
Graphs and Periods of the Trigonometric Functions We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let P = ( x , y ) be a point on the unit circle and let θ be the corresponding angle . Since the angle θ and θ +2 π correspond to the same point P , the values of the trigonometric functions at θ and at θ +2 π are the same. Consequently, the trigonometric functions are periodic functions. The period of a function f is defined to be the smallest positive value p such that f ( x + p ) = f ( x ) for all values x in the domain of f . The sine, cosine, secant, and cosecant functions have a period of 2 π . Since the tangent and cotangent functions repeat on an interval of length π , their period is π ( Figure 1.34 ).
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