Identities The set of real numbers for which an equation is defined is called the domain of the equation. If an equation is true for all values in its domain it is called an identity. Example 1. The equation has domain equal to all real numbers except 1. Example 2. The equation has domain equal to all real numbers except those of the form Since it is true for all numbers in its domain, it is an identity. Is the equation in Example 1 an identity?

Conditional equations If an equation is only true for some values in its domain, it is called a conditional equation. Example. The equation 2x +1 = 3 has domain equal to all real numbers, but it is only true for x = 1. Therefore, it is a conditional equation. Problem. Which equation is conditional?

Fundamental identities Reciprocal identities Quotient identities Pythagorean identities

Fundamental identities, continued Cofunction identities Even/Odd identities

Two possible ways to describe a right triangle

Using fundamental identities--an example Use fundamental identities to verify the trig identity We simplify the left-side of the identity.

Guidelines for verifying trigonometric identities Work with one side of the equation at a time. Look to factor an expression, add fractions, square a binomial, or create a monomial denominator. Look to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. If the preceding guidelines do not help, then try converting all terms to sines and cosines. Always try something.

Practice problems--verify the identity

Solving trigonometric equations If you are given a trigonometric equation, your task is to manipulate it using algebra and the fundamental trigonometric identities until you can write an equation of the form Example. Solve 2∙cos x = 1. Clearly, this is equivalent to cos x = 1/2. To solve for x, note that there are two solutions in [0, 2), namely /3 and 5/3. Also, because cos x has a period of 2, there are infinitely many other solutions, which can be written as Since the latter solution gives all possible solutions of the equation, it is called the general solution.

Graphical approaches to solving cos x = 1/2 The graph indicates how an infinite number of solutions can occur. Also, the unit circle shows infinitely many solutions occur.      

Solving trigonometric equations--an example Solve (tan 3x)(tan x) = tan 3x. This equation can be rewritten as (tan 3x)(tan x – 1) = 0. Next, set the factors to zero. We have To solve (i) for 3x in the interval [0, ), we have 3x = 0. In general, we have 3x = n so that x = n/3, n an integer. To solve (ii) for x in the interval [0, ), we have x = /4. In general, we have x = /4 + n, n an integer.

Solving trigonometric equations--another example Solve csc x + cot x = 1. First, let's convert to sines and cosines. We obtain Since there is no obvious way to solve 1 + cos x = sin x directly, we will try squaring both sides in the hope that the Pythagorean identity will result in a simplification. Of course, we know that squaring may introduce extraneous solutions. We have After canceling 1, the latter equation becomes cos x = 0 yields /2 + 2n, since 3/2 +2n is extraneous, cos x = –1 yields  + 2n, which is extraneous since csc x and cot x are undefined at x = .

Solving trigonometric equations--using inverse functions Solve The general solution is:

Guidelines for solving trigonometric equations Try to isolate the trigonometric function on one side of the equation. Look to use standard techniques such as collecting like terms and factoring (or use the quadratic formula). Look to use the fundamental identities. To solve equations that contain forms such as sin kx or cos kx, first solve for kx and then divide by k. If you can't get a solution using exact values, use inverse trigonometric functions to solve.

Practice problems--solving trigonometric equations

Application of sum and difference formulas

Application of sum and difference formulas--two examples Find the exact value of Verify that

Double-angle and power-reducing formulas Double-Angle Formulas Power-Reducing Formulas (double-angle formulas restated)

Using double-angle formulas to solve an equation Solve cos 2x + cos x = 0. First use the double-angle formula for cos 2x. n an integer

Using the power-reducing formulas Rewrite sin4x in terms of first powers of the cosines of multiple angles.

Other formulas not covered Half-Angle formulas, Product-to-Sum Formulas, and Sum-to-Product Formulas are interesting, but will not be covered in this course.

More practice problems--verify using all available identities

More practice problems--solve for x using all available identities

Solution of number 7 from previous slide Let u = x/2, then becomes Using a double angle formula, we have This becomes which factors as We have two cases (i) sin u = −1/2, (ii) sin u = 1. (i) yields u = 7π/6, u = 11π/6, (ii) yields u = π/2. Thus x = 7π/3, x = 11π/3, x = π , which are all the solutions in [0, 4π).