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Math-2 Honors Lesson 12.2 Fundamental Identities What you’ll learn about • • • • • • • Identities Basic Trigonometric Identities Pythagorean Identities Negative Angle Identities Even/Odd Identities Simplifying Trigonometric Expressions Solving Trigonometric Equations … and why Identities are important when working with trigonometric functions in calculus. Trigonometric Functions SOHCAHTOA “Some old horse caught another horse taking oats away.” opposite sin A hypotenuse o sin A h adjacent cos A hypotenuse a cos A h opposite tan A adjacent tan A o a SOH CAH TOA Trigonometric Functions Cosecant ratio hypotenuse(length) csc A opposite(length) h csc A o o sin A h 1 csc A sin A Trigonometric Functions Secant ratio hypotenuse(length) sec A adjacent (length) h sec A a a cos A h sec A 1 cos A Trigonometric Functions Cotangent ratio adjacent (length) cot A opposite (length) a cot A o o tan A a Shot your cow: 1 cot A tan A “Sha – Cho – Cao” h sec A a h csc A o a cot A o Trigonometric Functions Shot your cow: “Sha – Cho – Cao” h sec A a sin csc cos sec h csc A o a cot A o Unfortunately, ‘s’ doesn’t match up with ‘s’ (or ‘c’ with ‘c’) SHA-CHO-CAO SOHCAHTOA SOH “shot your cow” o sin A h CAH a cos A h TOA tan A 1 csc A 1 csc A sin A sin A h sec A a h o CHO a cot A o CAO csc A o a SHA cos A 1 sec A 1 tan A cot A sec A 1 cos A 1 cot A tan A What is an Identity? 2(x – 3) = 2x – 6 x 1 x 1 x 1 2 This is an equivalent expression. It is true for all real numbers. Like the equation above, it is a true statement. BUT, it is only true if x is in the domain of the expression on the right AND left side. When x = -1, it is not true. Identity: an equation that is true for all values that are in the domain of both sides of the equation. Identity: an equation that is true for all values that are in the domain of both sides of the equation. Is it an Identity? 1. 3x 2 6x 4 3x 2 2(3x 2) NOT an identity 2x 4x 2. 2x x2 2 x ( x 2) 2x x2 2 2x 2x Domain: left side: x ≠ 2 right side: all real #’s NOT an identity Basic Trigonometric Identities Reciprocal Identities 1 sin csc 1 cos sec 1 tan cot 1 csc sin 1 sec cos 1 cot tan Quotient Identities sin tan cos cos cot sin Find the Product sec Acos A h sec A a a cos A h hyp adj sec A cos A * adj hyp 1 Find the Product/quotient 3. tan Acos A 4. cot Asin A 5. tan A sin A 6. cot A sec A * * csc A sin A h sec A a h csc A o a cot A o a cos A h o sin A h o tan A a Using the UNIT CIRCLE (x, y) opp y sin y hyp 1 sin y r=1 y x Sine of the angle is the vertical distance from the x-axis to the point on the circle. Using the UNIT CIRCLE (x, y) r=1 y adj x x cos 1 hyp cos x x Cosine of the angle is the horizontal distance from the y-axis to the point on the circle. Using the UNIT CIRCLE The ordered pair (x, y) can be re-written as: (x, y) (cos , sin ) r=1 x y sin y cos x Using the UNIT CIRCLE Using Pythagorean Theorem: (cos , sin ) a2 b2 c2 x 2 y 2 12 (cos ) 2 (sin ) 2 1 r=1 x y We usually write (cos ) 2 as: cos 2 This give us our 1st “Pythagorean” identity. cos sin 1 2 2 Pythagorean Identities cos 2 sin 2 1 Divide both sides of the equation by cos 2 sin 2 cos 2 1 2 2 cos cos cos 2 This give us our 2nd “Pythagorean” identity. 1 tan sec 2 2 Your turn: 3. Divide the first identity by sin 2 and simplify to find the 3rd Pythagorean Identity. 1 cot csc 2 2 Solving equations 3x 1 5 To solve an equation we would use properties of equality to “isolate the variable” on one side of the equal sign. When using identities to solve equations we use substitution. Using Identities 3 Find sin if cos 5 Which of these identities will help? cos sin 1 2 2 cot 2 1 csc 2 1 tan sec 2 2 Substitution step. 2 3 2 sin 1 5 9 sin 2 1 25 9 sin 1 25 2 25 9 sin 25 25 2 25 9 sin 25 2 16 sin 25 4 sin 5 2 Your Turn: Given: 1 sin 2 7. cos Find: 2 Given: cos 2 8. Find: sin Using Identities Find sin and cos if tan = 4 Which of these identities will help? cos 2 sin 2 1 cot 2 1 csc 2 1 tan sec 2 2 Substitution step. 1 4 sec 2 2 1 sec cos 2 2 1 17 cos 2 1 cos 17 2 1 cos 17 17 cos 17 Using Identities Find sin and cos if tan = 4 17 cos 17 sin tan cos Which of these identities will help? 1 sec cos csc 1 sin cot cos sin Substitution step. sin 4 17 17 17 4* sin 17 4 17 sin 17 Your Turn: Given: sec(x) = 4, 9. 10. Find: tan(x) Find: cot(x) y Angle A: sin A r x cos A r x Angle B: sin B r y cos B r y tan A x x cot A y x tan B y y cot B x r sec A x r csc A y r sec B y r csc B x function of angle A = “cofunction” of angle B. Negative Angle Identities (“odd-even” identities) Sin θ = y coordinate of the point on the circle. sin(-θ) = -sin(θ) (x, y) “the y coord. of point through which (-θ) passes is the negative of the y-coord of the point through which (θ) passes. θ -θ (x, -y) Even-Odd Identities cos θ = x coordinate of the point on the circle. (x, y) Cos (-θ) = cos (θ) “the x coord. of point through which (-θ) passes is the same as the x-coord of the point through which (θ) passes. θ -θ (x, -y) Even-Odd Identities (x, y) Sin (-θ) = -sin (θ) 1 csc sin csc (-θ) = - csc (θ) θ -θ Cos (-θ) = cos (θ) sec (-θ) = sec (θ) (x, -y) 1 sec cos Even-Odd Identities Sin (-θ) = -sin (θ) sin tan cos Cos (-θ) = cos (θ) (x, y) tan( ) sin( ) cos( ) sin( ) tan( ) cos( ) θ -θ tan (-θ) = - tan (θ) cot (-θ) = - cot (θ) (x, -y) 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 The book uses ‘x’ instead of ‘θ’ for the angle variable. Find: sin( ) sec x -x 1 sin( ) cos( ) tan sin( ) sec sin( ) cos( ) Your Turn: 11. sec( ) cos ? 12. tan( ) cos ? Simplifying by Factoring and Using Identities Simplify: cot x sin x sec x tan x cos x csc x 3 3 Try converting tan, cot, sec, csc into functions of sin and cos cos 1 sin x 1 3 3 sin x cos x sin x cos x cos x sin x cos x sin 2 x From the properties of exponents: From the inverse Property of multiplication: From the Pythagorean Property 1 1 sin x cos 2 x cos x sin x sin 2 x cos 2 x 1 13. Simplify: Your turn: cot x tan x cos x cot x csc x cot x tan x cos x cos x 1 * * 1 1 sin x sin x 2 cos x 1 2 sin x cot 2 1 Pythagorean Identity 1 cot csc 2 csc x 2 2 Simplifying expressions with Identities sin 2 x, cos 2 x, tan 2 x, cot 2 x, sec 2 x, or csc 2 x Anytime you see refer to the Pythagorean Identities. 1 cos x sin x 2 sin 2 x sin x cos sin 1 2 2 1 tan sec 2 2 1 cot csc 2 2 sin x sin 1 cos 2 Use substitution 2 Your turn: use identities to simplify 2 1 tan x 13. 2 csc x 1 tan sec 2 sec 2 x 2 csc x 2 1 sin x 2 cos x 1 tan 2 x 2 14. sec x tan x sin x Convert into “sines” and “cosines” 1 sin x 1 * cos x sin x cos x 1 cos x sin sin x sin x * x cos x sin x 1 sin 2 x cos x sin x cos 2 x cos x sin x cos sin 1 2 Need common denominator! 2 cos x sin x cot x cos 1 sin 2 2 15. tan x tan x 2 2 csc x sec x sin x sin * cos x 1 2 Convert into “sines” and “cosines” sin x cos x * 1 cos x 2 sin x(sin cos x) cos x 2 sin x * (1) cos x 2 sin x cos x cos sin 1 2 tan x 2