Chapter 5 Analytic Trigonometry Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental
trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions.
I. Introduction (Page 352) Name four ways in which the fundamental trigonometric identities can be used: 1) to evaluate trigonometric functions
2) to simplify trigonometric expressions
3) to develop additional trigonometric identities
4) to solve trigonometric equations
The Fundamental Trigonometric Identities List six reciprocal identities: 1) sin u = 1/(csc u) 2) cos u = 1/(sec u) 3) tan u = 1/(cot u) 4) csc u = 1/(sin u) 5) sec u = 1/(cos u) 6) cot u = 1/(tan u) List two quotient identities: 1) tan u = (sin u)/(cos u) 2) cot u = (cos u)/(sin u) List three Pythagorean identities: 1) sin2 u + cos2 u = 1 2) 1 + tan2 u = sec2 u 3) 1 + cot2 u = csc2 u
List six cofunction identities: 1) sin(π/2 − u) = cos u 2) cos(π/2 − u) = sin u 3) tan(π/2 − u) = cot u 4) cot(π/2 − u) = tan u 5) sec(π/2 − u) = csc u 6) csc(π/2 − u) = sec u List six even/odd identities: 1) sin(− u) = − sin u 2) cos(− u) = cos u 3) tan(− u) = − tan u 4) csc(− u) = − csc u 5) sec(− u) = sec u 6) cot(− u) = − cot u
Course Number Instructor Date
What you should learn How to recognize and write the fundamental trigonometric identities
II. Using the Fundamental Identities (Pages 352−356) Example 1: Explain how to use the fundamental trigonometric
identities to find the value of tan u given that 2sec =u .
Use the Pythagorean identity 1 + tan2 u = sec2 u.
Substitute 2 for the value of sec u and solve for tan u.
Example 2: Explain how to use the fundamental trigonometric
identities to simplify xxx sintansec − . 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.
Example 3: Explain how to use a graphing utility to verify
whether xxxxx tancossinsinsec 3 =+ is an identity.
Graph y1 = sec x sin3 x + sin x cos x and y2 = tan x
in the same viewing window. If the two graphs appear to coincide, the expressions appear to be equivalent and the equation is an identity. If the two graphs do not coincide, then the equation is not an identity.
What you should learn How to use the funda-mental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions
Section 5.2 Verifying Trigonometric Identities Objective: In this lesson you learned how to verify trigonometric
identities. I. Introduction (Page 360) The key to both verifying identities and solving equations is . . .
the ability to use the fundamental identities and the rules of
algebra to rewrite trigonometric expressions.
An identity is . . . an equation that is true for all real values
in the domain of the variable.
II. Verifying Trigonometric Identities (Pages 360−364) Complete the following list of guidelines 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 a 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
and cotangents.
4) If the preceding guidelines do not help, try converting all
terms to sines and cosines.
5) Always try something! Even making an attempt that leads to
a dead end provides insight.
Example 1: Describe a strategy for verifying the identity
θθθθ seccostansin =+ . Then verify the identity.
Begin by converting all terms to sines and cosines.
Course Number Instructor Date
What you should learn How to understand the difference between conditional equations and identities
What you should learn How to verify trigonometric identities
Section 5.3 Solving Trigonometric Equations Objective: In this lesson you learned how to use standard algebraic
techniques and inverse trigonometric functions to solve trigonometric equations.
I. Introduction (Pages 368−370) To solve a trigonometric equation, . . . use standard
algebraic techniques such as collecting like terms and factoring.
The preliminary goal in solving trigonometric equations is . . .
to isolate the trigonometric function involved in the equation.
How many solutions does the equation 2sec =x 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. Example 1: Solve 01cos2 2 =−x . x = π/4 + nπ, x = 3π/4 + nπ 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.
II. Equations of Quadratic Type (Pages 370−372) Give an example of a trigonometric equation of quadratic type. Answers will vary. For example, cos2 x + 4 cos x + 4 = 0. To solve a trigonometric equation of quadratic type, . . .
factor the quadratic, or if factoring is not possible, use the
Quadratic Formula.
Course Number Instructor Date
What you should learn How to use standard algebraic techniques to solve trigonometric equations
What you should learn How to solve trigonometric equations of quadratic type
Example 2: Solve 1tan2tan 2 −=+ xx . x = 3π/4 + nπ Care must be taken when squaring each side 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.
III. Functions Involving Multiple Angles (Page 373) Give an example of a trigonometric function of multiple angles. Answers will vary. For example, tan 4x.
Example 3: Solve 224sin =x .
x = π/16 + nπ/2 and x = 3π/16 + nπ/2 IV. Using Inverse Functions (Page 374−375) Example 4: Use inverse functions to solve the equation
tan2 x + 4 tan x + 4 = 0. x = arctan (− 2) + nπ
Homework Assignment Page(s) Exercises
What you should learn How to solve trigonometric equations involving multiple angles
What you should learn How to use inverse trigonometric functions to solve trigonometric equations
Section 5.4 Sum and Difference Formulas Objective: In this lesson you learned how to use sum and difference
formulas to rewrite and evaluate trigonometric functions. I. Using Sum and Difference Formulas (Pages 380−383) List the sum and difference formulas for sine, cosine, and tangent. sin(u + v) = sin u cos v + cos u sin v
sin(u − v) = sin u cos v − cos u sin v
cos(u + v) = cos u cos v − sin u sin v
cos(u − v) = cos u cos v + sin u sin v
tan(u + v) = (tan u + tan v)/(1 − tan u tan v)
tan(u − v) = (tan u − tan v)/(1 + tan u tan v)
Example 1: Use a sum or difference formula to find the exact
value of tan 255°. (9 + 6√3 + 3)/6 Example 2: Find the exact value of cos 95° cos 35° + sin 95°
sin 35°. 1/2 A reduction formula is . . . a formula involving
expressions such as sin(θ + nπ/2) or cos(θ + nπ/2), where n is an
integer, that can be derived from sum and difference formulas.
Example 3: Derive a reduction formula for ⎟⎠⎞
⎜⎝⎛ +
2sin πt .
sin(t + π/2) = cos t
Course Number Instructor Date
What you should learn How to use sum and difference formulas to evaluate trigonometric functions, to verify identities, and to solve trigonometric equations
The power-reducing formulas are: sin2 u = (1 − cos 2u)/2 cos2 u = (1 + cos 2u)/2 tan2 u = (1 − cos 2u)/(1 + cos 2u) III. Half-Angle Formulas (Pages 390−391) List the half-angle formulas:
=2
sin u ± √(1 − cos u)/2
=2
cos u ± √(1 + cos u)/2
=2
tan u (1 − cos u)/(sin u) = (sin u)/(1 + cos u)
The signs of sin (u/2) and cos (u/2) depend on . . . the
quadrant in which u/2 lies.
Example 2: Find the exact value of tan 15°. 2 − √3 IV. Product-to-Sum Formulas (Pages 391−393) The product-to-sum formulas are used in calculus to . . .
evaluate integrals involving the products of sines and cosines of
two different angles.
The product-to-sum formulas are: sin u sin v = 1/2[cos(u − v) − cos (u + v)]
cos u cos v= 1/2[cos(u − v) + cos (u + v)]
sin u cos v = 1/2[sin(u + v) + sin(u − v)]
cos u sin v = 1/2[sin(u + v) − sin(u − v)]
What you should learn How to use half-angle formulas to rewrite and evaluate trigonometric functions
What you should learn How to use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions
Section 5.5 Multiple-Angle and Product-to-Sum Formulas 93
Example 3: Write cos 3x cos 2x as a sum or difference. 1/2 cos x + 1/2 cos 5x
The sum-to-product formulas can be used to . . . rewrite a
sum or difference of trigonometric functions as a product.
The sum-to-product formulas are: sin u + sin v = 2 sin((u + v)/2) cos((u − v)/2) sin u − sin v = 2 cos((u + v)/2) sin((u − v)/2) cos u + cos v = 2 cos((u + v)/2) cos((u − v)/2) cos u − cos v = − 2 sin((u + v)/2) sin((u − y)/2) Example 4: Write cos 4x + cos 2x as a sum or difference. 2 cos 3x cos x Additional notes
