Pre- Calculus Mathematics Trigonometric Identities and Equations

Transcription

1 Pre- Calculus Mathematics Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on a Trigonometric Expression The FUNdamental TRIGonometric Identities In trigonometry, there are expressions and equations that are true for any given angle. These are called identities. An infinite number of trigonometric identities exist, and we are going to prove many of these identities, but we are going to need some basic identities first. The six basic trig ratios will lead to our first identities sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = And our knowledge of Pythagoras will determine the remaining FUNdamental TRIGonometric Identities

2 Pre- Calculus Mathematics Trigonometric Identities and Equations Pythagorean Identities: sin θ + cos θ = tan θ = sec θ 1 + cot θ = csc θ Reciprocal and Quotient Identities: secθ = 1 cosθ cscθ = 1 sinθ cotθ = 1 tanθ tanθ = sinθ cosθ cotθ = cosθ sinθ Corollary Identities ( a statement that follows readily from a previous statement) sin θ + cos θ = tan θ = sec θ 1 + cot θ = csc θ Simplifying a Trigonometric Expression There are many different strategies to simplifying a trigonometric expression. The following examples will look at the most common types of strategies. Write as a fraction with a common denominator sin θ cos θ + cos θ Factor as a difference of squares 1 cos! θ sec! θ tan! θ Change everything to sine and cosine. tan θ sec θ Multiply by the conjugate 1 1 sin θ

3 Pre- Calculus Mathematics Trigonometric Identities and Equations Now we can use these strategies along with the eight fundamental identities to simplify expressions Example 1: Simplify sin θ + cos! θ sin θ Example 2: Simplify 1 sin! θ 1 Example 3: Simplify!!!"!!!"#!!!"#! Example 4: Simplify cos! x 2cos! x + 1 Example 5: Simplify!"#!!!!"#!!!"#!!"#! Example 6: Simplify!"#!!!!"#! +!"#!!!!"#!

4 Pre- Calculus Mathematics Trigonometric Identities and Equations Restrictions Just like any algebraic expression, a trigonometric expression cannot have zero in the denominator. We must consider the exact values that would result in a denominator of zero. Example 7: Determine the restrictions on tan x csc x for 0 x < 2π Practice: Page 264 #2-9 (as many as needed)

5 Pre- Calculus Mathematics Verifying Trigonometric Identities Goal: Verify and prove Trigonometric Identities Trigonometric Identities When verifying trigonometric identities, the key is using the rules for algebra as well as the fundamental trigonometric identities to rewrite and simplify expressions. An identity has been proven when the right side of the equal sign is the same as the left side of the equal side. Example 1: Prove the identity:!!!"#!!!"#!! = tan! θ Example 2: Prove the identity:!!!"#!!"#! = sin θ + cos θ

6 Pre- Calculus Mathematics Verifying Trigonometric Identities Example 3: Prove the identity:!"#$!!"#$!"#!!! = cot θ

7 Pre- Calculus Mathematics Verifying Trigonometric Identities Example 4: Prove the identity: sec θ csc θ tan θ = sec! θ Practice: Page 271 #1-26

8 Pre- Calculus Mathematics Trigonometric Equations Goal: 1. Solve trigonometric equations for conditional statements and general form 2. Solve equation with angles other then θ or x. Trigonometric Equations A trigonometric equation is different from a trigonometric identity in that the equation is true for some values of the variable and not all values. When solving a trig equation, we can solve for a specified domain of values ( usually 0 θ < 2π ) or in general form ( for all potential values) Example 1: Solve: sec θ =!! for a) 0 θ < 2π b) general form Example 2: Solve: tan θ = for a) 0 θ < 2π b) general form Example 3: Solve: 2 cos! x cos x 1 = 0 for a) 0 x < 2π b) general form

9 Solving trigonometric equations with angles other than θ Pre- Calculus Mathematics Trigonometric Equations Solving a trig equation for angles with coefficients ( 2θ, 3θ, etc) follows very similar steps: 1. Isolate the trig function. eg. sin 2θ 2. Determine the general solution(s) for angle 2θ 3. Determine the general solution(s) for angle θ. (divide by the coefficient) 4. Determine the specific ( conditional ) solution(s) for the given interval Example 4: Solve: sin 2x = 1 for a) general form b) 0 x < 2π Example 5: Solve: 3 tan 2θ + 1 = 0 for a) general form b) 0 θ < 2π Example 6: Solve: 4 sin! 2x + 2 sin 2x 2 = 0 for a) 0 θ < 2π b) general form Practice: Page 281 #1-5

10 Pre- Calculus Mathematics Sum and Difference Identities Goal: 1. Use sum and difference identities to solve complex trig problems 2. Simplify expressions and prove identities involving sums and differences Sum and Difference Identities Identities are not limited to the fundamental identities and single angles. We can also use identities involving sums and differences. The derivations of these identities are shown on page 287 of your textbook. We will be looking not at proving these identities but using these identities. cos( α +β ) = cos αcosβ sin αsin β ( ) cos( α β ) = cos αcosβ+ sin αsin β ( ) tan ( α ) tan ( α ) sin α+β = sin αcos β+ cos αsinβ sin α β = sin αcosβ cos αsinβ +β = tan α + tan β β = tan α tan β 1 tan αtan β 1 + tan αtan β Example 1: Find the exact value: Example 2: Find the exact value: cos 345 sin!! cos!! + cos!! sin!! Example 3: Express as a single function, then evaluate. cos!!! sin!!! Example 4: Express as a single function, then evaluate.!"#!!!"#!!!!!"#!!"#!!

11 Pre- Calculus Mathematics Sum and Difference Identities Example 5: Given angle A in quadrant I and angle B in quadrant II, such that sin A =!! find tan A B. and cos B =!"!", Example 6: Prove:!"#!!!"#!!!! = tan x Practice: Page 292 #1-6

12 Pre- Calculus Mathematics Double- Angle Identities Goal: 1. Identify the double angle trigonometric identities 2. Simplify and prove double- angle trigonometric expressions and equations Double Angle Identities Using the sum and difference identities, we can determine other trigonometric identities sin ( α+β ) = sin αcos β+ cos αsinβ cos( α +β ) = cos αcosβ sin αsin β tan ( α ) +β = tan α + tan β 1 tan αtan β Double Angle Identities (same angle) sin 2θ = 2sinθ cosθ cos 2 cos sin θ = θ θ = 2 2cos θ 1 2 = 1 2sin θ tan 2θ =!!"#!!!!"#!!

13 Pre- Calculus Mathematics Double- Angle Identities Example 1: Simplify 2 1 cos4x Example 2: Solve, 0 x 2π csc! x = 2 sec 2x Example 3: Prove:!!!"#!!!"#!! =!!!"#!!"#!!!"#! Practice: Page 300 #1 6 (as many as needed)