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Questions and Answers
Which of the following represents a horizontal translation of the cosine function that results in an equivalent expression?
Which of the following represents a horizontal translation of the cosine function that results in an equivalent expression?
What is the exact value of tan(5π/3)?
What is the exact value of tan(5π/3)?
What type of symmetry does the function y = cos θ exhibit?
What type of symmetry does the function y = cos θ exhibit?
Using the addition formula for sine, what is sin(105°) expressed in terms of sine and cosine of other angles?
Using the addition formula for sine, what is sin(105°) expressed in terms of sine and cosine of other angles?
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Which of the following equations is equivalent to sin(θ)?
Which of the following equations is equivalent to sin(θ)?
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What is the formula for cos(a + b) in terms of sin and cos?
What is the formula for cos(a + b) in terms of sin and cos?
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What is the equivalent expression for cos(5π/3 + 2π)?
What is the equivalent expression for cos(5π/3 + 2π)?
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If sin x = -4/5 and sin y = -12/13, what is cos(x+y)?
If sin x = -4/5 and sin y = -12/13, what is cos(x+y)?
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Which function is NOT considered an odd function?
Which function is NOT considered an odd function?
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Which of the following expressions uses complementary angles to depict an equivalent trigonometric expression?
Which of the following expressions uses complementary angles to depict an equivalent trigonometric expression?
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Which expression represents the double angle formula for sine?
Which expression represents the double angle formula for sine?
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Which of the following is NOT a form of the cosine double angle formula?
Which of the following is NOT a form of the cosine double angle formula?
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The equation sin(θ) = sin(θ + 2π) is an example of which concept?
The equation sin(θ) = sin(θ + 2π) is an example of which concept?
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Which of the following transformations reflects the function y = sin θ across the x-axis?
Which of the following transformations reflects the function y = sin θ across the x-axis?
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To evaluate sin2θ given cosθ = -4/5 and π/2 ≤ θ ≤ π, what is the value?
To evaluate sin2θ given cosθ = -4/5 and π/2 ≤ θ ≤ π, what is the value?
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What is the form of the tangent double angle formula?
What is the form of the tangent double angle formula?
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What is the value of sin(π/3)?
What is the value of sin(π/3)?
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Which identity represents the relationship between sine and cosine in reciprocal identities?
Which identity represents the relationship between sine and cosine in reciprocal identities?
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Which relationship describes the connection between sin and cos for complementary angles?
Which relationship describes the connection between sin and cos for complementary angles?
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Which one of the following is a correct Pythagorean identity?
Which one of the following is a correct Pythagorean identity?
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If θ = π/8, what is the complementary angle?
If θ = π/8, what is the complementary angle?
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Which formula represents the addition formula for cosine?
Which formula represents the addition formula for cosine?
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In Quadrant II, what is the value of cos(π - θ)?
In Quadrant II, what is the value of cos(π - θ)?
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What is the double angle formula for sine?
What is the double angle formula for sine?
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What happens to the sign of the sine function in Quadrant III?
What happens to the sign of the sine function in Quadrant III?
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Which of the following is a cofunction identity?
Which of the following is a cofunction identity?
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For the angle 3π/8, what is the complementary ratio of csc(3π/8)?
For the angle 3π/8, what is the complementary ratio of csc(3π/8)?
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Which statement about tan(π + θ) is true in Quadrant III?
Which statement about tan(π + θ) is true in Quadrant III?
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What is the value of cos(π/2 + x)?
What is the value of cos(π/2 + x)?
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Given the equation cos 2x = -1/2, what quadrants should you consider for the solutions?
Given the equation cos 2x = -1/2, what quadrants should you consider for the solutions?
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How many solutions does sin 4x = 0 yield on the interval 0 ≤ x ≤ 2π?
How many solutions does sin 4x = 0 yield on the interval 0 ≤ x ≤ 2π?
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What is the first step to solve the equation -5 cos x + 3 = 2?
What is the first step to solve the equation -5 cos x + 3 = 2?
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What is the related acute angle B when cos B = 1/5?
What is the related acute angle B when cos B = 1/5?
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For the function cos(x - y) / cos(x + y), what is the simplified form?
For the function cos(x - y) / cos(x + y), what is the simplified form?
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In which interval does cos x = 1/5 yield valid solutions?
In which interval does cos x = 1/5 yield valid solutions?
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What is the total number of solutions for cos 2x = 1 within the interval 0 ≤ x ≤ 2π?
What is the total number of solutions for cos 2x = 1 within the interval 0 ≤ x ≤ 2π?
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What are all possible solutions for the equation $2x = \pi - \frac{\pi}{3}$?
What are all possible solutions for the equation $2x = \pi - \frac{\pi}{3}$?
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How is the depth $d(t)$ of the water at Matthews Cove represented mathematically?
How is the depth $d(t)$ of the water at Matthews Cove represented mathematically?
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What angles correspond to the solutions when solving $\tan 2x = 1$?
What angles correspond to the solutions when solving $\tan 2x = 1$?
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In the equation $2\sin x \cos x = \cos 2x$, what trigonometric identity is applied to simplify the left side?
In the equation $2\sin x \cos x = \cos 2x$, what trigonometric identity is applied to simplify the left side?
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What approach can be utilized to solve a quadratic trigonometric equation?
What approach can be utilized to solve a quadratic trigonometric equation?
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At what time will the water depth first reach 2 meters in Matthews Cove?
At what time will the water depth first reach 2 meters in Matthews Cove?
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Which of the following is true regarding the solutions for $\sin²x - \sin x - 2 = 0$?
Which of the following is true regarding the solutions for $\sin²x - \sin x - 2 = 0$?
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What are the values of $t$ when the depth $d(t)$ is 2 meters during the first 12 hours?
What are the values of $t$ when the depth $d(t)$ is 2 meters during the first 12 hours?
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Study Notes
Trigonometric Identities and Equations
- Trigonometric functions are periodic, leading to multiple equivalent expressions.
- Equivalent expressions can be found using horizontal translations, symmetry, complementary angles, and related acute angles.
- Trigonometric functions can be shifted horizontally to create equivalent expressions
- Sine and cosine functions have relationships based on their complementary angles, where sin(x) = cos(90-x).
- Even functions exhibit symmetry about the y-axis (f(x) = f(-x)).
- Odd functions show symmetry about the origin (f(-x) = -f(x)).
- Horizontal translations of sine or cosine functions by multiples of the period result in equivalent expressions.
- The CAST rule relates trigonometric ratios across quadrants.
- Compound angle formulas allow for adding or subtracting angles to obtain exact trigonometric values.
Compound Angle Formulas
- Addition formulas: sin(a + b) = sin a cos b + cos 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)
- Subtraction formulas: sin(a - b) = sin a cos b - cos 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)
Double Angle Formulas
- Sine: sin(2θ) = 2 sin θ cos θ
- Cosine: cos(2θ) = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
- Tangent: tan(2θ) = (2 tan θ)/(1 - tan² θ)
Solving Trigonometric Equations
- Trigonometric equations frequently have infinite solutions due to periodicity.
- A specific interval (often 0 ≤ θ ≤ 2π) usually confines the domain.
- Techniques include factoring, the quadratic formula, and applying trigonometric identities (Pythagorean, double angle, etc.)
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Description
Test your knowledge on trigonometric identities and equations with this quiz. Explore the relationships between sine and cosine functions, discover compound angle formulas, and understand how periodic functions create equivalent expressions. Are you ready to challenge yourself on these essential mathematical concepts?
