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Questions and Answers
What defines an identity in trigonometric equations?
What defines an identity in trigonometric equations?
Which technique is NOT suggested for verifying trigonometric identities?
Which technique is NOT suggested for verifying trigonometric identities?
If an equation does not satisfy some values in its domain, what is it classified as?
If an equation does not satisfy some values in its domain, what is it classified as?
Which of the following is NOT a fundamental identity type mentioned?
Which of the following is NOT a fundamental identity type mentioned?
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When verifying trigonometric identities, which strategy is recommended?
When verifying trigonometric identities, which strategy is recommended?
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What is the radian measure of a 90 degree angle?
What is the radian measure of a 90 degree angle?
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What is the sine of 180 degrees?
What is the sine of 180 degrees?
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What are the coordinates of the point on the unit circle that corresponds to an angle of 3π/4 radians?
What are the coordinates of the point on the unit circle that corresponds to an angle of 3π/4 radians?
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What is the tangent of 270 degrees?
What is the tangent of 270 degrees?
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Which of the following angles has a cosine value of -1/2?
Which of the following angles has a cosine value of -1/2?
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What is the radian measure of 30 degrees?
What is the radian measure of 30 degrees?
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Study Notes
Trigonometric Identities
- Trigonometric identities are equations that are true for all values of the variable in the domain of the equation
- An equation that is not an identity is called a conditional equation. This occurs when some values of the variable in the domain of the equation do not satisfy the equation
Fundamental Trigonometric Identities
-
Reciprocal Identities
- sin x = 1/csc x
- cos x = 1/sec x
- tan x = 1/cot x
- csc x = 1/sin x
- sec x = 1/cos x
- cot x = 1/tan x
-
Quotient Identities
- tan x = sin x/cos x
- cot x = cos x/sin x
-
Pythagorean Identities
- sin² x + cos² x = 1
- 1 + tan² x = sec² x
- 1 + cot² x = csc² x
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
Guidelines for Verifying Trigonometric Identities
- Work with each side of the equation independently. Start with the more complicated side and work towards the other side in a series of steps.
- Analyze the identity and look for opportunities to apply fundamental identities.
- Techniques to utilize include:
- Rewrite the more complicated side in terms of sines and cosines
- Factor out the greatest common factor
- Separate a single-term quotient into two terms
- Combine fractional expressions using the least common denominator
- Multiply the numerator and the denominator by a binomial factor that appears on the other side of the identity
Additional Trigonometric Identities
-
Sum and Difference Identities
- sin(a + b) = sin a cos b + cos a sin b
- sin(a - b) = sin a cos b - cos a sin b
- cos(a + b) = cos a cos b - sin a sin b
- cos(a - b) = cos a cos b + sin a sin b
- tan(a + b) = (tan a + tan b) / (1 - tan a tan b)
- tan(a - b) = (tan a - tan b) / (1 + tan a tan b)
-
Double Angle Identities
- sin 2a = 2 sin a cos a
- cos 2a = cos² a - sin² a = 2 cos² a - 1 = 1 - 2 sin² a
- tan 2a = 2 tan a / (1 - tan² a)
-
Half Angle Identities
- sin (a/2) = ±√((1-cos a)/2)
- cos (a/2) = ±√((1+cos a)/2)
- tan (a/2) = ±√((1 - cos a) / (1 + cos a)) = sin a / (1 + cos a) = (1 - cos a) / sin a
Example Identities to Verify
- csc θ cot θ = sin θ / (1 + cos θ)
- sec x cot x = csc x
- cos² x - cos x sin² x = cos³ x
- (1 + sin x) / cos x = sec x + tan x
Seatwork Examples (Identities to Prove)
- 2 sin² θ = 2 - 2 cos² θ
- csc θ sec θ - cot θ = tan θ
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Description
Test your knowledge of trigonometric identities with this quiz. Explore various reciprocal, quotient, and Pythagorean identities as well as even-odd identities. Understand the key concepts that are crucial for solving trigonometric equations.
