Identities and Equations An equation such as y 2 – 9y + 20 = (y – 4)(y – 5) is an identity...
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Transcript of Identities and Equations An equation such as y 2 – 9y + 20 = (y – 4)(y – 5) is an identity...
![Page 1: Identities and Equations An equation such as y 2 – 9y + 20 = (y – 4)(y – 5) is an identity because the left-hand side (LHS) is equal to the right-hand.](https://reader030.fdocuments.in/reader030/viewer/2022032600/56649db35503460f94aa29fe/html5/thumbnails/1.jpg)
TRIGONOMETRIC IDENTITIES
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Identities and Equations
An equation such as y2 – 9y + 20 = (y – 4)(y – 5) is an identity because the left-hand side (LHS) is equal to the right-hand side (RHS) for whatever value is substituted to the variable.
Based on the example, an identity is defined as an equation, which is true for all values in the domain of the variable.
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Identities and Equations
There are identities which involve trigonometric functions. These identities are called trigonometric identities.
Trigonometric identity is an equation that involves trigonometric functions, which is true for all the values of θ for which the functions are defined.
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Identities and Equations
A conditional equation is an equation that is true only for certain values of the variable.
The equations y2 – 5y + 6 = 0 and x2 – x – 6 = 0 are both conditional equations. The first equation is true only if y = 2 and y = 3 and the second equation is true only if x = 3 and x = -2.
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The Fundamental Identities
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The Fundamental Identities
Reciprocal Identities
Reciprocal Identities Equivalent Forms Domain Restrictions
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Quotient (or Ratio) Identities
Quotient Identities Domain Restrictions
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Pythagorean Identities
Negative Arguments Identities
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Notes:
The real number x or θ in these identities may be changed by other angles such as α, β, γ, A, B, C,….
The resulting identities may then be called trigonometric identities.
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Example:
Find the remaining circular functions of θ using the fundamental identities, given sin θ = and P(θ) ϵ II.
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Simplifying Expressions
Examples:
Simplify the following expressions using the fundamental identities.
1. tan3 x csc3 x
2. sec x • cos x – cos2 x
3. (csc2 x – 1)(sec2 x sin2 x)
4.
5. (cos θ – 1)(cos θ + 1)
6. sin2 θ + cot2 θ sin2 θ
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Proving Identities There is no exact procedure to be followed in
proving identities. However, it may be helpful to express all the given functions in terms of sines and cosines and then simplify.
To establish an identity, we may use one of the following:
1. Transform the left member into the exact form of the right.
2. Transform the right into the exact form of the left, or
3. Transform each side separately into the same form.
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Examples
1. Prove that + = is an identity.
2. Verify if tan2 β – sin2 β = tan2 β sin2 β is an identity.
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Exercises
1. If sin θ = and P(θ) is in quadrant IV, find the other trigonometric function values of θ using the fundamental identities.
2. Express cos θ (tan θ – sec θ) in terms of sine and cosine using the fundamental identities and then simplify the expression.
3. Show that (1 + cot2 θ)= is an identity.
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Exercises
4. Simplify the following expressions.
a) (sec x + tan x)(sec x – tan x)
b) 2 –
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Do Worksheet 6
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Sum and Difference Identities
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Double-Angle Identities
Sine Cosine Tangent
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Half-Angle Identities
Sine Cosine Tangent
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Product-to-Sum and Sum-to-Product Identities
Product-to-Sum Identities
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Product-to-Sum and Sum-to-Product Identities
Sum-to-Product Identities
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Examples 1. Find the exact value of sin 75° using sum
and difference identities.
2. Simplify sin 20°cos 40° + cos 20°sin 40°.
3. Simplify tan(x + 4π).
4. Given that cot θ = and θ is in the second quadrant, find:
a) sin 2θ b) tan 2θ c) cos 2θ
5. Find the exact value of sin 22.5° using half-angle identities.
6. Simplify cot (90° - θ) if tan θ = using cofunction identities.
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7. Evaluate tan 165°.
8. Find the exact value of 105°.
9. Simplify cot ( - x ).
10. Simplify the following trigonometric expressions:
a) b)