Algebra 2/Trig AIIT.17 Trig Identities Notes. Name: Date: Block:

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1 Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Name: Date: Block: Trigonometric Identities When two trig expressions can be proven to be equal to each other, the statement is called a trig identity Example of a trig identity o Recall: in the unit circle, x = r cos θ and y = r sin θ tan y x r sin sin tan r cos cos This is our first trig identity! Reciprocal Identities We already know these! Reciprocal Identities sin θ = csc cos θ = sec cot csc θ = sin sec θ = cos tan Quotient Identities We ve already proven one of them: What can we conclude for cot θ? sin tan cos Quotient Identities sin cos cos sin

2 Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Page Pythagorean Identities By the Pythagorean Theorem: x + y = r In the circle, what does x =? In the circle, what does y =? Substitute: Simplify: From this basic identity, we can derive others sin θ + cos θ = Divide both sides by sin θ: Divide both sides by cos θ: Pythagorean Identities sin θ + cos θ = + tan θ = sec θ + cot θ = csc θ Cofunction Identities Suppose we have the triangle at right: We know: o θ + φ are complementary o θ + φ = 90 o θ = 90 o φ and φ = 90 o - θ 3 3 o sin θ =, cos φ = 5 5 sin θ = cos φ sin θ = cos (90 o θ) and cos φ = sin (90 o φ) Cofunctions are complementary!

3 Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Page 3 We can apply the same steps to the other trig functions, which leads to three sets of cofunction identities: sin tan sec Cofunction Identities = cos θ cos = cot θ cot = csc θ csc = sin θ = tan θ = sec θ Even/Odd Identities (Negative Angle Identities) Odd/Even Functions: o Odd: f(-x) = -f(x) Example: y = x 3 o Even: f(-x) = f(x) Example: y = x o Odd trig functions: sin, tan, cot, csc o Even trig functions: cos, sec o Use observation of graphs to show sin is odd and cos is even o Prove remaining trig functions even or odd: f(-x) = -f(x) (odd functions) f(-x) = -f(x) (even functions) tan(-x) = sec(-x) = cot(-x) = csc(-x)= Odd/Even 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) SO WHAT? Use trig identities to evaluate trig functions Use trig identities to simplify expressions Use trig identities to verify other trig identities

4 Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Page 4 4 Example : Given sin θ = and, find the values of the other trig values of 5 θ. First find cos θ: sin θ + cos θ = (Pythagorean identity) Find the other trig values: Example : Simplify sin cot Replace cotθ with its quotient identify: Cancel: Simplified form: Example 3: Simplify tan sin Example 4: Simplify csc cot sin Example 5: Simplify cos A sec A tan A You Try ) Find the values of the other five trig functions of θ. 3 3 a) cos, 0 b) sin, 6 7 ) Simplify the expressions: tan xcsc x a) sin xcot xsec x b) sec x c) cos sin( ) d) tan( ) cos e) sin cos cot

5 Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Page 5 Verifying Trig Identities Create new trig identities from the ones we ve already proven Start with an expression on one side until you ve manipulated it algebraically to be identical to the other sec Example 6: Verify the identity sin sec Work on expression on the left: Example 7: Verify the identity cos x sec x tan x sin x Example 8: Word problem! A vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the sun above the horizon is θ can be modeled by the equation below. Show that the equation is equivalent to s hcot. sin( 90 ) s h sin You Try: Verify the identities a) csc ( sin x) cot x b) cos x csc xtan x c) (tan x )(cos x ) tan x d) sec cos sin tan 4 4 e) sin (tan cot ) sec f) sin x cos x sin x

6 Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Page 6 Summary of Identities Reciprocal Identities sin θ = csc cos θ = sec cot csc θ = sin sec θ = cos tan sin cos Quotient Identities cos sin Pythagorean Identities sin θ + cos θ = + tan θ = sec θ + cot θ = csc θ sin tan sec Cofunction Identities = cos θ cos = cot θ cot = csc θ csc = sin θ = tan θ = sec θ Odd/Even 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)