5.1-5.3 Math Analysis Test

sin²x + cos²s=

1

Guidlines for Verifying Trigonometric Identities

1. Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2. Look for opportunities to factor an expression, add fractions, square binomial, or create a monomial denominator. 3. Look for opportunities 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 cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Always try something. Even paths that lead to dead ends provide insight.

Describe a strategy for verifying the identity 2 2 sin (csc −1)(csc +1) =1− sin . Then verify the identity.

Because the left side is more complicated, start with it. Begin by multiplying (csc x − 1) by (csc x + 1), and then search for a fundamental identity that can be used to replace the result.

Tricks when proving Trig Identities

Conjugate, factoring, and common denominator

To prove Trig Identities you need to...

Make sides match by taking one side and simplifying till you get the other side. parts of work along right side of eq sign straight down.

Explain how to use the fundamental trigonometric identities to simplify sec x − tan x sin x .

Rewrite the expression in terms of sines and cosines. Combine the resulting fractions to obtain (1 − sin2 x)/(cos x). Using the Pythagorean identity sin2 u + cos2 u = 1, replace the numerator with cos2 x. Simplify the result to obtain cos x.

How many solutions does the equation sec x = 2 have? Explain.

The equation has an infinite number of solutions because the secant function has a period of 2π. Any angles coterminal with the equation's solutions on [0, 2π) will also be solutions of the equation. however for tan it is [0,π)

The Fundamental Identities

There are four groups of fundamental identities: reciprocal identities, quotient identities, Pythagorean identities, and even-odd identities. The fundamental identities are used to establish other relationships among trigonometric functions.

Explain how to use the fundamental trigonometric identities to find the value of tan u given that sec u = 2 .

Use the Pythagorean identity 1 + tan2u = sec2u. Substitute 2 for the value of sec u and solve for tan u.

To solve an equation in which two or more trigonometric functions occur

collect all terms on one side and try to separate the functions by factoring or by using appropriate identities.

sinx=

cos(∏/2 -x)

tanx=

cot(∏/2 -x)

cot²x + 1=

csc²x

To solve a trigonometric equation of quadratic type

factor the quadratic, or if this is not possible, use the Quadratic Formula.

cscx=

sec(∏/2 -x)

tan²x + 1=

sec²x

Reciprocal Identities

sin x = 1 / csc x cos x = 1 / sec x tan x = 1 / cot x csc x = 1 / sin x sec x = 1 / cos x cot x = 1 / tan x

Even-Odd Identities

sin(-x) = - sin x cos(-x) = cos x tan (-x) = - tan x csc (-x) = - csc x sec (-x) = sec x cot (-x) = - cot x

cosx=

sin(∏/2 -x)

Pythagorean Identities

sin²x + cos²x = 1 1 + tan²x = sec²x 1 + cot²x = csc²x

Quotient Identities

tan x = sin x / cos x cot x = cos x / sin x

The key to verifying identities is

the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions.

You cant divide by trig identities and get rid of them because

then you get rid of possible solutions

Care must be taken when squaring both sides of a trigonometric equation to obtain a quadratic because

this procedure can introduce extraneous solutions, so any solutions must be checked in the original equation to see whether they are valid or extraneous

The preliminary goal in solving trigonometric equations is

to isolate the trigonometric function involved in the equation

To solve a trigonometric equation(5.3)

use standard algebraic techniques such as collecting like terms and factoring.