Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
PreCalculus Section 5-1 Trigonometric Identities Name:___________________ Basic Trig Identites: See key concept box on page 312 Reciprocal Identities 1 csc 1 csc sin sin 1 sec 1 sec cos Quotient Identities 1 cot 1 cot tan cos tan sin cos cos cot sin tan Ex. 1: Use the trig identities to find the trig values. a. If cos θ = ¾ , find sec θ. b. If sec x = 5 3 and tan x = , find sin x. 4 4 You can use the unit circle to define trig functions. For any angle θ, sine and cosine are the legs of a right triangle and the hypotenuse is 1. This leads to the Pythagorean identities. The signs depend on which quadrant the triangle lies in. See key concept on page 313. Pythagorean Identities sin cos 1 2 2 tan 2 1 sec2 Ex. 2: a. If cot θ = 2 and cos θ < 0, find sin θ and cos θ. b. If sin x 1 , cos x > 0, find cot x and sec x. 6 PreCalculus Ch 5 Notes_Page 1 cot 2 1 csc2 A trig function f is a cofunction of trig function g if f(α) = g(β) when α and β are complementary angles. Look at triangle on top of page 314 and the key concept boxes. Cofunction Identities sin cos 2 tan cot 2 sec csc 2 cos sin 2 cot tan 2 csc sec 2 Odd Even Identities sin sin cos cos tan tan csc csc sec sec cot cot Ex. 4: Use the Odd-Even and Cofunction Identities to find the following. a. If cos x = -0.75, find sin x . 2 b. If sin x = -0.37, find cos x . 2 To simplify a trig expression, start by rewriting it in terms of one trig function or in terms of sine and cosine only. Ex. 4: Simplify. 1 1 sin 2 x a. cos x PreCalculus Ch 5 Notes_Page 2 b. sec x – tan x sin x Ex. 5: Simplify by applying identities and factoring. a. cos xtan x – sin xcos2 x Ex. 6: Simplify by combining fractions. sec x sec x a. 1 sec x 1 sec x b. b. csc x tan 2 x sec x 2 cos x 1 sin x 1 sin x cos x Ex. 7: Rewrite the expression as an expression that does not involve a fraction. (Hint: use conjugates) 1 tan 2 x cos 2 x a. b. csc2 x 1 sin x PreCalculus Ch 5 Notes_Page 3 PreCalculus Section 5-2 Verifying Trigonometric Identities Name:___________________ To verify an identity means to prove that both side of an equation are equal for all values of the variable for which both sides are defined. Start with the more complicated side and work toward the other side. You will need state a reason or justification for each step (do a proof). Ex. 1: Verify each of the following. tan 2 x 1 a. sec4 x 2 1 sin x b. sec2 cot 2 1 cot 2 When there are multiple fractions with different denominators, you can find a common denominator to reduce the expression to one fraction. Ex. 2: Verify. 2 cot x PreCalculus Ch 5 Notes_Page 4 sin x sin x 1 cos x 1 cos x When the denominator is in the form 1±u or u±1, use the conjugate of the denominator. Ex. 3: Verify. sin x cos x cot x cot x sec x 1 Until an identity has been verified, you can’t assume that both sides of the equation are equal. Therefore, you can’t use properties of equality to perform algebraic operations on each side of an identity. When the more complicated expression of an identity involves powers, try factoring. Sometimes, it’s helpful to work out each side and meet somewhere in the middle. Ex. 4: Verify. a. cos x sec2 x tan x cos x tan 3 x sin x PreCalculus Ch 5 Notes_Page 5 b. cot 3 x cot x cos x csc3 x * Strategies for verifying trig identities: see Key Concept Summary box on page 323. Start with the more complicated side of the identity and work to transform it into the simpler side, keeping the other side of the identity in mind as your goal. Use reciprocal, quotient, Pythagorean, and other basic trig identities. Use algebraic operations such as combining fractions, rewriting fractions as sums and differences, multiplying expressions, or factoring expressions. Work each side separately to reach a common intermediate expression. If no other strategy presents itself, try converting the entire expression to one involving only sines and cosines. Identifying Identities and Nonidentities: You can use a graphing calculator to investigate whether an equation may be an identity by graphing the functions related to each side of the equation. A graphing calculator may confirm an identity, but it does not prove an identity. You still need to provide the algebraic verification of an identity. Ex. 5: Use a graphing calculator to test whether each equation is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both side are defined, but not equal. 1 tan 2 x a. tan x csc x sec x b. cos3 x sin 3 x cos 2 x sin 2 x cos x sin x PreCalculus Ch 5 Notes_Page 6 PreCalculus Section 5-3 Solving Trigonometric Equations Name:___________________ See the diagram on page 327 to show that there are an infinite number of solutions for trig functions. You need to add integer multiples of the period to get all of the solutions. When there is only one trig expression, start by isolating this expression. Ex. 1: Solve. a. 5cos x 3cos x 3 b. 4sin x 2sin x 2 Ex. 2: Solve by taking the square root of each side. (see study tip on page 328) a. 3 tan 2 x 4 3 b. 3cot 2 x 4 7 PreCalculus Ch 5 Notes_Page 7 You may need to solve by factoring or by using the Quadratic Formula. Ex. 3: Find all solutions of each equation on the interval [0, 2π). a. 2sin x cos x 3 sin x b. 2sin 2 x sin x 1 0 When equations involve multiple angles, such as cos 2x= 1, first solve for the multiple angle 2x. Ex. 4: A projectile is sent off with an initial speed v0 of 350m/s and clears a fence 3000m away. The height of the fence is the same height as the initial height of the projectile. If the distance the projectile v 2 sin 2 traveled is given by d 0 , find the interval of possible launch angles to clear the fence. 9.8 PreCalculus Ch 5 Notes_Page 8 Ex. 5: Find all solutions of sin 2 x sin x 1 cos 2 x on the interval [0, 2π). Sometimes you can square both sides of the equation, but this may produce extraneous solutions. So make sure to check for the extraneous solutions when you squared the original equation. Ex. 6: Find all solutions of sin x cos x 1 on the interval [0, 2π). PreCalculus Ch 5 Notes_Page 9 PreCalculus Sections 5-4 Sum and Difference Identities Name:___________________ Look at page 336 to see the Cosine of a Sum and Difference Identities. See the key concept on page 337. Sum Identities Difference Identities cos cos cos sin sin cos cos cos sin sin sin sin cos cos sin sin sin cos cos sin tan tan tan 1 tan tan tan tan tan 1 tan tan You can rewrite angle measures as the sums and differences of special angles and use these identities to find exact values of trig functions of angles that are less common. Ex. 1: Find the exact value of each trig expression. a. cos 75˚ b. tan 11 12 Ex. 2: An alternating current i in amperes in a certain circuit can be found after t seconds using i= 4(sin255)t. (a) Rewrite the formula in terms of the sum of two angle measures. (b) Use a sine of a sum identity to find the exact current after 1 second. PreCalculus Ch 5 Notes_Page 10 If a trig expression has the form of a sum or difference identity, you can use the identity to find the exact value or to simplify an expression by rewriting the expression as a function of a single angle. Ex. 3: a. Find the exact value of tan 78 tan18 1 tan 78 tan18 b. Simplify: sin 3 cos 4 cos 3 sin Sum and Difference Identities can be used to rewrite trig expressions as algebraic expressions. 3 arccos x as an algebraic expression of x that does not involve trig Ex. 4: Write cos arcsin 2 functions. Ex. 5: Verify: cos cos PreCalculus Ch 5 Notes_Page 11 4 Sum and difference identities can be used to rewrite trig expressions in which one of the angles is a multiple of 90˚ or π/2 radians. The resulting identity is called a reduction identity because it reduces the complexity of the expression. Ex. 6: Verify each reduction identity. a. cos sin 2 b. tan x 360 tan x Ex. 7: Find the solutions of sin x sin x 0 on the interval [0, 2π). 4 4 PreCalculus Ch 5 Notes_Page 12 PreCalculus Section 5-5 Multiple Angle Identity Name:_________________________ By letting α and β both equal θ in each of the angle sum identities, you can derive the following double-angle identities. Double Angle Identities sin 2 2sin cos tan 2 2 tan 1 tan 2 cos 2 cos 2 sin 2 cos 2 2 cos 2 1 cos 2 1 2sin 2 Ex. 1: a. If sin 3 on the interval 0, , find sin2θ, cos2θ, and tan2θ. 4 2 3 b. If cos on the interval 0, , find sin2θ, cos2θ, and tan2θ. 5 2 Ex. 2: Solve cos2θ – cosθ = 2 on the interval [0, 2π). PreCalculus Ch 5 Notes_Page 13 The double-angle identities can be used to derive the following power-reducing identities. Power Reducing Identities sin 2 1 cos 2 2 cos 2 1 cos 2 2 tan 2 1 cos 2 1 cos 2 Ex. 3: Rewrite the trig function in terms of cosines of multiple angles with no power greater than 1. a. csc 4 b. cos 4 Ex. 4: Solve sin 2 cos 2 cos 0 PreCalculus Ch 5 Notes_Page 14 in each of the power-reducing identities, you can derive the half-angle identities. 2 The sign of the identity that involves the ± is determined by which quadrant the terminal side of lies. By replacing θ with 2 Half Angle Identities sin tan 2 2 1 cos 1 cos 1 cos 2 tan cos 2 1 cos sin Ex. 5: Find the exact values. a. sin 22.5˚ b. tan 2 1 cos 2 tan 2 7 12 Ex. 6: Solve the trig equations on the interval [0, 2π]. x x x a. sin 2 cos 2 b. 2sin 2 cos x 1 sin x 2 2 2 PreCalculus Ch 5 Notes_Page 15 sin 1 cos