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8.3 TRIGONOMETRIC FUNCTIONS: RELATIONSHIPS AND GRAPHS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Relationships Between the Graphs of Sine and Cosine Example 1 Use the fact that the graphs of sine and cosine are horizontal shifts of each other to find relationships between the sine and cosine functions. Solution The figure suggests that the y = sin t graph of y = sin t is the graph of y = cos t y = cos t shifted right by π/2 radians (or by 90◦). Likewise, the graph of y = cos t is the graph of y = sin t shifted left by π /2 radians. Thus, sin t = cos(t − π/2) cos t = sin(t + π/2) The graph of y = sin t can be obtained by shifting the graph of y = cos t to the right by π/2 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Relationships Involving The Tangent Function sin tan cos for cos 0 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Relationships Involving Reciprocals of the Trigonometric Functions The reciprocals of the trigonometric functions are given special names. Where the denominators are not equal to zero, we define secant θ = sec θ = 1/cos θ. cosecant θ = csc θ = 1/sin θ. cotangent θ = cot θ = 1/tan θ = cos θ/sin θ. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Relationships Between the Graphs of Secant and Cosine Example 3 Use a graph of g(θ) = cos θ to explain the shape of the graph of f(θ) = sec θ. Solution The figure shows the graphs of cos θ and sec θ. In the first quadrant cos θ decreases from 1 to 0, so sec θ increases from 1 toward +∞. The values of cos θ are negative in the second quadrant and decrease from 0 to −1, so the values of g(θ) = cos θ sec θ increase from −∞ to −1. The graph of y = cos θ is symmetric about the vertical line θ = π, so the graph of f(θ) = f(θ) = sec θ sec θ is symmetric about the same line. Since sec θ is undefined wherever cos θ = 0, the graph of f(θ) = sec θ has vertical asymptotes at θ = π/2 and θ = 3π/2. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Relationships Between the Graphs of Secant and Cosine The graphs of y = csc θ and y = cot θ are obtained in a similar fashion from the graphs of y = sin θ and y = tan θ, respectively. Plots of y = csc θ and y = sin θ Plots of y = cot θ and y = tan θ Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally The Pythagorean Identity We now see an extremely important relationship between sine and cosine. The figure suggests that no matter what the value of θ, the coordinates of the corresponding point P satisfy the following condition: x2 + y2 = 1. But since x = cos θ and y = sin θ, this means cos2 θ + sin2 θ = 1 y 1 θ x ● P = (x, y) = (cos θ, sin θ) y x Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Summarizing the Trigonometric Relationships • Sine and Cosine functions sin t = cos(t − π/2) = −sin(−t) cos t = sin(t + π/2) = cos(−t) • Pythagorean Identity cos2 θ + sin2 θ = 1 • Tangent and Cotangent tan θ = cos θ/sin θ and cot θ = 1/tan θ • Secant and Cosecant sec θ = 1/cos θ and csc θ = 1/sin θ Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally