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LESSON 5 Section 6.3. Trig Functions of Real Numbers. UNIT CIRCLE Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.

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## LESSON 5 Section 6.3

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**LESSON 5 Section 6.3**Trig Functions of Real Numbers**UNIT CIRCLE**Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle. APPENDIX IV of your textbook shows a good unit circle.**This is a second revolution around the unit circle. This is**another ‘period’ of the curve.**y = sin x**• This is a periodic function. The period is 2π. • The domain of the function is all real numbers. • The range of the function is [-1, 1]. • It is a continuous function. The graph is shown on the next slide.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Graphing the sine curve for -2π ≤ x ≤ 2π. (π/2, 1) (2π, 0) (π, 0) (0, 0) (3π/2, - 1)**UNIT CIRCLE**Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.**Make a table of x and y values for y = cos x**Remember, the y value in this table is actually the x value on the unit circle.**y = cos x**• This is a periodic function. The period is 2π. • The domain of the function is all real numbers. • The range of the function is [-1, 1]. • It is a continuous function. The graph is shown on the next slide.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Graphing the cosine curve for -2π ≤ x ≤ 2π. (0, 1) (2π, 1) (π/2, 0) (3π/2, 0) (π, - 1)**How do the graphs of the sine function and the cosine**function compare? • They are basically the same ‘shape’. • They have the same domain and range. • They have the same period. • If you begin at –π/2 on the cosine curve, you have the sine curve.**The notation above is interpreted as: ‘as x approaches the**number π/6 from the right (from values of x larger than π/6), what function value is sin x approaching?’ Since the sine curve is continuous (no breaks or jumps), the answer will be equal to exactly the sin (π/6) or ½ . The notation below is interpreted as: ‘as x approaches the number π/6 from the left (from values of x smaller than π/6), what function value is sin x approaching?’ Again, since the sine curve is continuous, the answer will be equal to exactly the sin (π/6) or ½ .**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. Use the graph to verify these values.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. Q I Q IV**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.Use the graph to verify these values.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. Q IV Q III**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.**Make a table of x and y values for y = tan x**Remember, tan x is (sinx / cosx).**y = tanx**• This is a periodic function. The period is π. • The domain of the function is all real numbers, except those of the form π/2 +nπ. • The range of the function is all real numbers. • It is not a continuous function. The function is undefined at -3π/2, -π/2, π/2, 3π/2, etc. There are vertical asymptotes at these values. The graph is shown on the next slide.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Graphing the tangent curve for -2π ≤ x ≤ 2π. (π/4, 1) (-π/4, -1)**For all x values where the tangent curve is continuous,**approaching from the left or the right will equal the value of the tangent at x. However, the two cases above are different; because there is a vertical asymptote when x = -π/2. If approaching from the left (the smaller side), the answer is infinity. If approaching from the right (the larger side), the answer is negative infinity.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. tan x = 1 Q I Q III**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation. Q I Q III**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation. Q IV Q II**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.**Sketch the graph of y = sin x + 1**This will be a graph of the basic sine function, but shifted one unit up. The domain will be all real numbers. What would be the range? Since the range of a basic sine function is [-1, 1], the domain of the function above would be [0, 2].**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Sketch the graph of y = sin x + 1**Sketch the graph of y = cos x - 2**This would be the graph of a basic cosine function shifted 2 units down. The domain is still all real numbers. What is the range? The basic cosine function has a range of [-1, 1]. The range of the function above would be [-3, -1].**π**-π -π 2 π 2 -3π 2 2π 3π 2 -2π Sketch the graph of y = cos x - 2**Find the intervals from –2π to 2πwhere the graph of y =**tan x is: • Increasing • Decreasing • Remember: No brackets should be used on values of x where the function is not defined. • Increasing: [-2π, -3π/2) b) The function never decreases. • (-3π/2, -π/2) • (-π/2, π/2) • (π/2, 3π/2) • (3π/2, 2π]

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