Mathematics
Grade 11
15 min
Complementary angle identities
Complementary angle identities
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1
Introduction & Learning Objectives
Learning Objectives
Define complementary angles and identify cofunction pairs.
Recall and write the complementary angle identities in both degrees and radians.
Apply complementary angle identities to simplify trigonometric expressions.
Solve trigonometric equations by utilizing cofunction relationships.
Prove more complex trigonometric identities that involve complementary angles.
Evaluate trigonometric functions of angles greater than 90° by relating them to their complementary acute angles.
Ever noticed how the sine of 30° is the exact same as the cosine of 60°? 🤔 This isn't a coincidence; it's a fundamental property of right-angled triangles!
This tutorial explores the relationship between trigonometric functions of complementary angles (angles that add up to...
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Key Concepts & Vocabulary
TermDefinitionExample
Complementary AnglesTwo angles are complementary if their sum is 90° (or π/2 radians). In a right-angled triangle, the two acute angles are always complementary.30° and 60° are complementary angles because 30° + 60° = 90°. Similarly, π/6 and π/3 are complementary because π/6 + π/3 = 3π/6 = π/2.
Trigonometric IdentityAn equation involving trigonometric functions that is true for all values of the variable for which both sides of the equation are defined.The Pythagorean identity, sin²(θ) + cos²(θ) = 1, is true for any angle θ.
CofunctionsPairs of trigonometric functions where the value of one function for a given angle is equal to the value of the other function for the complementary angle. The 'co' in cosine, cotangent, and cosecant stands for 'compleme...
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Core Formulas
Sine and Cosine Cofunction Identity
sin(90° - θ) = cos(θ) OR sin(π/2 - θ) = cos(θ)
cos(90° - θ) = sin(θ) OR cos(π/2 - θ) = sin(θ)
Use this to switch between sine and cosine. The sine of an angle is the cosine of its complement, and vice versa.
Tangent and Cotangent Cofunction Identity
tan(90° - θ) = cot(θ) OR tan(π/2 - θ) = cot(θ)
cot(90° - θ) = tan(θ) OR cot(π/2 - θ) = tan(θ)
Use this to switch between tangent and cotangent. The tangent of an angle is the cotangent of its complement, and vice versa.
Secant and Cosecant Cofunction Identity
sec(90° - θ) = csc(θ) OR sec(π/2 - θ) = csc(θ)
csc(90° - θ) = sec(θ) OR csc(π/2 - θ) = sec(θ)
Use this to switch between secant and cosecant. The secant of an angle is the cosecant of its complement, and vice versa.
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Challenging
Given that sin(θ) = 5/13 and cos(α) = 5/13, where θ and α are acute angles, find the value of sin(α).
A.12/13
B.5/13
C.13/12
D.5/12
Challenging
To prove the identity tan(θ) + cot(θ) = sec(θ)csc(θ), a student rewrites the left side as (sin(θ)/cos(θ)) + (cos(θ)/sin(θ)). What is the most effective next step that utilizes a complementary angle relationship implicitly?
A.Multiply the expression by tan(90°-θ).
B.Combine the fractions to get (sin²(θ) + cos²(θ))/(sin(θ)cos(θ)).
C.Substitute θ with (90°-α).
D.Convert everything to secant and cosecant.
Challenging
If f(x) and g(x) are a pair of trigonometric cofunctions (e.g., f=sin, g=cos) and f(A) = g(B) for acute angles A and B, which of the following equations must be true?
A.A/B = 1
B.+ B = π
C.- B = 0
D.+ B = π/2
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