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1 Trig/AP Calc A Semester Version 0.. Created by James Feng fengerprints.weebly.com

2 Trig/AP Calc A - Semester Handy-dandy Identities Know these like the back of your hand. "But I don't know the back of my hand!" It's never too late to start learning.. Reciprocal Identities sinθ = cosθ = tanθ = cscθ secθ cotθ cscθ = secθ = cotθ = sinθ cosθ tanθ.2 Pythagorean Identities sin 2 Θ + cos 2 Θ = tan 2 Θ + = sec 2 Θ + cot 2 Θ = csc 2 Θ.3 Cofunction Identities sina = cos(90 A) seca = csc(90 A) tana = cot(90 A) cosa = sin(90 A) csca = sec(90 A) cota = tan(90 A).4 Negative-Angle Identities sin( Θ) = sinθ csc( Θ) = cscθ cos( Θ) = cosθ sec( Θ) = secθ tan( Θ) = tanθ cot( Θ) = cotθ

3 Trig/AP Calc A - Semester 2.5 Cosine Sum and Difference Identities cos(a + B) = cosacosb sinasinb cos(a B) = cosacosb + sinasinb.6 Sine Sum and Difference Identities sin(a + B) = sinacosb + cosasinb sin(a B) = sinacosb cosasinb.7 Tangent Sum and Difference Identities tan(a + B) = tan(a B) = tana + tanb tanatanb tana tanb + tanatanb.8 Double-Angle Identities cos2a = cos 2 A sin 2 A cos2a = 2sin 2 A cos2a = 2cos 2 A sin2a = 2sinAcosA tan2a = 2tanA tan 2 A = sin2θ cos2θ.9 Half-Angle Identities cos A 2 = ± + cosa 2

4 Trig/AP Calc A - Semester 3.9 Half-Angle Identities (cont) sin A 2 = ± cosa 2 tan A 2 = ± cosa + cosa = sina + cosa = cosa sina -.9 Quotient Identities tanθ = sinθ cosθ cotθ = cosθ sinθ 2 The Unit Circle Although you don't necessarily have to memorize this, it's a great tool for solving equations, determining quadrants, etc.

5 Trig/AP Calc A - Semester 4 2 Unit Circle (cont) Remember: The cosine of a value on the unit circle is the x value that corresponds with the degree/radian measure. The sine of a value on the unit circle is the y value that corresponds with the degree/radian measure. The tangent of a value on the unit circle is the y value over the x value (y/x, see quotient identities above). 3 Range of Trigonometric Functions Alright, we get it. You made a stupid mistake because you forgot about these. DON'T DO IT AGAIN. 3. Graphs of Inverse Functions From left to right: Top row: arcsin(x) or sin - (x) Quadrants: I and IV arccos(x) or cos - (x) Quadrants: I and II arctan(x) or tan - (x) Quadrants: I and IV y -π/2 or π/2 Bottom row: arcsec(x) or sec - (x) Quadrants: I and II y π/2 arccsc(x) or csc - (x) Quadrants: I and II y 0 arccot(x) or cot - (x) Quadrants: I and IV y 0 or π

6 Trig/AP Calc A - Semester Graphs of Sin, Cos, and Tan Functions and their Inverses 4 Radians An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of radian. 4. Converting with Degrees (Degree measure) (Radian measure) 80 π 60 would convert to π/3 radians. π radians would convert to 80. π = Radian measure 80 = Degree measure See the unit circle above for more common conversions.

7 Trig/AP Calc A - Semester Applications of Radian Measure Arc Length - The length s of an arc intercepted on a circle of radius r by a central angle of measure Θ radians is given by the product of the radius and the radian measure of the angle, or: s = rθ Area of a Sector - The area of a sector of a circle of radius r and central angle Θ is given by: A = 2 r2 Θ Θ r s 4.3 Linear and Angular Speed Angular speed/velocity (w) Linear speed/velocity (v) w = Θ t v = s t v = rθ t v = rw Note: For the pulley problems, make sure you remember that v = v2. This means that the linear velocity for both wheels is the same! 5 Translations This will be a very brief summary of how translations affect the graphs of the trigonometric functions. Note that the equation for a sine/cosine graph is: y = c + a sin b (x - d) or y = c + a cos b (x - d), where a is the amplitude, b is the period, c is the vertical shift, and d is the phase shift. Amplitude = half the difference between the maximum and minimum values The graph of y = a sin x or y = a cos x, with a 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except with range [- a, a ]. The amplitude is a. In red, or the "shorter" one: y = sin x In blue, or the "taller" one: y = 2 sin x Notice how the 2 doubles the range of the equation. The range is now [-2,2] instead of [-,].

8 Trig/AP Calc A - Semester 7 5 Translations (cont) Period = the length of one cycle The period is not affected by the amplitude. No matter what the amplitude, the periods of y = a sin x and y = a cos x will be 2π. For b > 0, the graph of y = sin bx will resemble that of y = sin x, but with period 2π/b. This is the same for y = cos bx. In red, or the "wider" one: y = cos x In blue, or the "narrower" one: y = cos 2x With a period change, the 2 makes the length of one period smaller. Using 2π/b (2π/2), we determine that the length of one period for the blue function is π units. From the book: Guidelines for Sketching Graphs of Sine and Cosine Functions. Find an interval whose length is one period 2π/b by solving the three-part inequality 0 b(x - d) 2π. 2. Divide the interval into four equal parts (The standard five are 0, π/2, π, 3π/2, and 2π). 3. Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c. 4. Plot the points found in Step 3, and join them with a curve having amplitude a. 5. Draw the graph over additional periods, to the right and to the left, as needed. Phase Shifts A phase shift is another name for a horizontal translation. Use the inequality mentioned in Step above to determine the interval of one cycle of the graph, then follow the remaining steps to graph your function. Phase shifts are represented by d. Vertical Shifts A vertical shift is a constant that "centers" the graph on a horizontal line that is not the x-axis. Vertical shifts are represented by c.

6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function.

Math 160 www.timetodare.com Periods of trigonometric functions Definition A function y f ( t) f ( t p) f ( t) 6.4 & 6.5 Graphing Trigonometric Functions = is periodic if there is a positive number p such