Chapter 7, (js), Analytic Trigonometry

7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas State the Double-Angle Formulas for sin, cos, and tan.

pg 542

What are the Half-Angle Formulas for sin, cos, and tan?

pg 544

What are the formulas for lowering powers for sin, cos, tan?

pg 544

7.2 - Addition and Subtraction Formulas State the addition and subtraction formulas for sin, cos, and tan.

table pg 535

What is a trigonometric equation?

An equation that involves trigonometric functions, such as sinθ - 1/2 = 0

How do you approach expressions such as 1/2(sin x) + ((sqrt 3)/2)(cos x)?

538

7.5 - Trigonometric Equations What is a trigonometric equation? How do you solve them?

An equation that contains trigonometric functions. e.g., 2 sin x - 1 = 0. Solve by isolating the trig function.

Trig equation problems may ask for all solutions, or solutions in a certain interval. How do you approach if asked for all solutions?

First find solutions in the period for the function. e.g., -pi/2 to pi/2 for tan, then add the multiple of the period, k(pi) for tan.

7.4 - Inverse Trigonometric Functions How do you make each trig function one-to-one so that it can have an inverse?

For cos limit the domain from [0 to pi]. For sin limit the domain from [-pi/2 to pi/2]. For tan limit the domain from (-pi/2 to pi/2). For sec limit domain to 0 </= x < pi/2, and pi </= x < 3pi/2. For csc limit domain as follows: 0 < x </= pi/2 and pi < x </= 3pi/2. For cot limit domain as follows: 0 < x < pi.

How would you find cos (pi/12)? (Not using half-angle formula.)

Take cos(pi/4 - pi/6)

Try problems 41 and 43, pg 540. If no luck, see explanation on pg 538.

When doing these problems, start by plotting the point with coordinates equal to the coefficients in the expression. Next, calculate the hypotenuse. The coefficients divided by the hypotenuse give you the cos and sin of the 2nd angle.

State the Product-to-Sum and Sum-to-Product Formulas for sin and cos.

pgs 546, 547

7.1 - Trigonometric Identities What are the cofunction identities?

sin(pi/2 - u) = cos u cos(pi/2 - u) = sin u tan(pi/2 - u) = cot u cot(pi/2 - u) = tan u sec(pi/2 - u) = csc u csc(pi/2 - u) = sec u

How do you use Double-Angle Formulas when you have angles such as 4x, 6x, etc...

As with 2x, on the right side the angle is half what the angle is on the left of the formula. e.g., for sin 6x, you have 2(sin3x)(cos3x)

Find cos((sin^-1(3/5)) two different ways. Remember that the inverse of any trig function represents an angle θ. For example, cos^-1 x = θ. Another example, cos(2 tan^-1 x) = cos 2θ. In the first case, draw the triangle with angle θ, and then one side = x and the the hypotenuse is 1. In the second case, draw the triangle with angle θ, opposite side = x, and adjacent side = 1.

js 552

How do you approach an equation with a "Multiple Angle" such as 2(sin 3x) -1 = 0 (finding all solutions, and then solutions in [0, 2pi) interval)?

First find all solutions for the multiple angle 3x, then divide by 3. Next, since the interval is [0, 2pi) for x, it is [0, 6pi) for 3x, so find all solutions in 3 turns of unit circle, then divide by 3. js566 (need to find something that better explains the logic). Perhaps a better way to look at this is example 8 on js566. After finding all solutions, limit answers to those that are in the interval, eg, 29pi/18 is in the [0, 2pi) interval, but 37pi/18 is outside the interval.

How do you approach an equation such as 2cos^2(x) - 7cos(x) + 3 = 0

Factor to get 2 factors equal to zero, and solve. js563.