Transcript Chapter 7
5-Minute Check Lesson 7-1A Chapter 7 Trigonometric Identities and Equations Section 7.1 Basic Trigonometric Identities Definitions Identity β A statement of equality between two expressions that is true for all values of the variable(s) for which the expressions are defined. ex: π₯ 2 β π¦ 2 = π₯ β π¦ π₯ + π¦ Trigonometric Identity β is an identity involving trigonometric expressions. ex: sin π cos π = tan π Reciprocal functions β (We talked about this previouslyβ¦ this is just review) π¬π’π§ π½ = π ππ¬π π½ ππ¨π¬ π½ = π π¬ππ π½ π¬ππ π½ = π ππ¨π¬ π½ πππ§ π½ = π ππ¨π π½ ππ¨π π½ = π πππ§ π½ ππ¬π π½ = π π¬π’π§ π½ Opposite Angle Identities π¬π’π§(βπ½) = β π¬π’π§ π½ ππ¨π¬ βπ½ = ππ¨π¬ π½ Definitions continued Quotient Identities β π¬π’π§ π½ ππ¨π¬ π½ = πππ§ π½ ππ¨π¬ π½ π¬π’π§ π½ Pythagorean Identities β Recall with the unit circle we knew : π₯ 2 + π¦ 2 = 1 And we know : cos π = π₯, sin π = π¦ By substitution : πππ 2 π + π ππ2 π = 1 Which we write as: ππππ π½ + ππππ π½ = π If we divide both sides by sineβ¦ π + ππππ π½ = ππππ π½ If we divide both sides by cosineβ¦ ππππ π½ + π = ππππ π½ = ππ¨π π½ Examples Use the given information to find the trigonometric values 3 2 1. If sec π = , find cos π. 4 3 2. If csc π = , find tan π when cos π > 0 1 5 3. If sin π = β , find cos π when tan π < 0 Prove that each equation is not a trigonometric identify by producing a counterexample 1. sin π cos π = cot π 2. sec π tan π = sin π 3. sin π + cos π = 1π More Examples: Simplify: sin π₯ + sin π₯ πππ‘ 2 π₯ 1. Look for any GCFs: * π πππ₯ β sin π₯ (1 + πππ‘ 2 π₯) 2. Look for any identities: * 1 + πππ‘ 2 π₯ β sin π₯ (ππ π 2 π₯) sin π₯ ( 3. Change everything to sines and cosines 4. Simplify 5. Simplify 1 sin π₯ 1 ) π ππ2 π₯ csc π₯ Simplify: cos π₯ tan π₯ + sin π₯ cot π₯ Simplify: 1 + πππ‘ 2 π₯ β πππ 2 π₯ β πππ 2 π₯πππ‘ 2 π₯ THESE STEPS ARE NOT IN ANY ORDER. EACH PROBLEM IS SPECIAL AND YOU MUST OPEN YOUR MIND TO HOW TO SOLVE THEM. Your answers will always be 1 term, 1 number or a binomial left with sine and cosine only ο Homework: Page 427: #19 β 51 Odd, 57, 69 You Try It You try It Section 7.2 Verify Trigonometric Identities Suggestions for Verifying trigonometric Identities 1. 2. 3. 4. 5. Transform the more complicated side of the equation into the simplier side. Substitute one or more basic trigonometric identity to simplify expression. Factor or multiply to simplify Multiply expressions by an expression equal to 1. Express all trigonometric functions in terms of sine and cosine. Example: Verify that π ππ 2 π₯ β tan π₯ cot π₯ = π‘ππ2 π₯ 1 2 π ππ π₯ β tan π₯ = π‘ππ2 π₯ tan π₯ π ππ 2 π₯ β 1 = π‘ππ2 π₯ π‘ππ2 π₯ + 1 β 1 = π‘ππ2 π₯ π‘ππ2 π₯ = π‘ππ2 π₯ Lesson Overview 7-2A Lesson Overview 7-2B Lesson Overview 7-2C You Try It You Try It Section 7.3 Sum and Difference Identities Sum and Difference Identities Sum/Difference Identity for Sine sin(πΌ ± π½) = sin πΌ cos π½ ± sin π½ cos πΌ *SINE = SIGN SAME* Sum/Difference Identity for Cosine cos(πΌ ± π½) = cos πΌ cos π½ β sin πΌ sin π½ *COSINE = NO SIGN SAME* Sum/Difference Identity of Tangent tan πΌ ± tan π½ tan(πΌ ± π½) = 1 β tan πΌ tan π½ *TANGENT β SAME/DIFFERENT* Lesson Overview 7-3A Lesson Overview 7-3B Lesson Overview 7-3C Examples 1. cos 105° 2. sin 165° π 3. sin 12 4. tan 23π 12 5. sec 1275° π 6. Find the exact value if 0 < π₯ < and 0 < π¦ < 2 3 24 cos(π₯ β π¦) ππ cos π₯ = πππ cos π¦ = 5 25 π 2 Homework: Page 442: #15-31 Odd, 34-38 All, 40,42 5-Minute Check Lesson 7-4A 5-Minute Check Lesson 7-4B Lesson Overview 7-4A Lesson Overview 7-4B 5-Minute Check Lesson 7-5A 5-Minute Check Lesson 7-5B Lesson Overview 7-5A 5-Minute Check Lesson 7-6A Lesson Overview 7-6A Lesson Overview 7-6B 5-Minute Check Lesson 7-7A 5-Minute Check Lesson 7-7B Lesson Overview 7-7A Lesson Overview 7-7B