Podcast
Questions and Answers
What is the value of cos 30°?
What is the value of cos 30°?
$\frac{\sqrt{3}}{2}$
What is the argument of $2 = \log_3 9$?
What is the argument of $2 = \log_3 9$?
9
What is the reciprocal function of sin θ?
What is the reciprocal function of sin θ?
csc θ
In which quadrant is tan > 0 and sin < 0?
In which quadrant is tan > 0 and sin < 0?
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What is 90° in radians?
What is 90° in radians?
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What is the equation y = x + 4 shifted to y = x - 4?
What is the equation y = x + 4 shifted to y = x - 4?
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Is y = x^2 + 8 a one-to-one function?
Is y = x^2 + 8 a one-to-one function?
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Is $6^2 = 36$ correct according to the equation $2 = \log_6 36$?
Is $6^2 = 36$ correct according to the equation $2 = \log_6 36$?
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What is the value of cot θ in terms of adj/opp?
What is the value of cot θ in terms of adj/opp?
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What is the reference angle for 145°?
What is the reference angle for 145°?
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What is the expression for $\log \left( \frac{\sqrt{x}}{y} \right)$ in terms of a single logarithm?
What is the expression for $\log \left( \frac{\sqrt{x}}{y} \right)$ in terms of a single logarithm?
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What is the expression for $\log x + 2 \log y$?
What is the expression for $\log x + 2 \log y$?
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What is x in the equation $5^x = 625$?
What is x in the equation $5^x = 625$?
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What are the values for $\log x^2 + \log 2 = \log 8$?
What are the values for $\log x^2 + \log 2 = \log 8$?
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What is the value of tan 45°?
What is the value of tan 45°?
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What is the value of sin 330°?
What is the value of sin 330°?
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Solve for θ: $5 \tan θ = 4$.
Solve for θ: $5 \tan θ = 4$.
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Solve for θ: $2 \sin θ = -1$.
Solve for θ: $2 \sin θ = -1$.
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Solve for θ: $5 \cos θ = -3$.
Solve for θ: $5 \cos θ = -3$.
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Solve for θ: $2 \cos^2 θ = 1$.
Solve for θ: $2 \cos^2 θ = 1$.
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If bc = 1 and ef = 3 when ac = 2, what is df?
If bc = 1 and ef = 3 when ac = 2, what is df?
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If jk = 4 and mn = 10 when jl = 8, what is mo?
If jk = 4 and mn = 10 when jl = 8, what is mo?
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What is the value of cos 225°?
What is the value of cos 225°?
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What is the value of sin 225°?
What is the value of sin 225°?
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What is the value of tan 225°?
What is the value of tan 225°?
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Solve for c: $c^2 = 4^2 + 6^2 = \sqrt{52}$.
Solve for c: $c^2 = 4^2 + 6^2 = \sqrt{52}$.
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What does $8 \times 10^6 \times \log 10$ equal?
What does $8 \times 10^6 \times \log 10$ equal?
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Study Notes
Trigonometric Functions
- √3/2 represents the cosine of 30°, a key value in trigonometry.
- csc(θ) is the reciprocal of sin(θ).
- sin(330°) equals -1/2, indicating it is in the fourth quadrant.
- cos(225°) and sin(225°) both equal -√2/2, showing common values in the third quadrant.
- tan(45°) = 1, a fundamental angle in trigonometric calculations.
- tan > 0 and sin < 0 indicates such conditions exist in Quadrant III.
- Reference angle for 145° is 35°, derived from subtracting 180°.
Logarithmic Functions
- For log₃(9), the argument is 9, illustrating how logarithms express exponentiation (3^2).
- log(√(x/y)) simplifies to (1/2)log(x) - log(y), showing properties of logarithms.
- log(x) + 2 log(y) translates to log(xy²), providing another logarithmic identity.
- The equation log(x²) + log(2) = log(8) leads to possible solutions of x being ±2.
Changes in Variables
- The equations y = x + 4 and y = x - 4 show transformations of a linear function.
Quadratic Functions
- The function y = x² + 8 fails to be one-to-one since it is parabolic and symmetric.
Angle Relationships and Solutions
- Angles such as 30°, 330°, and 210° represent specific angle situations in trigonometric equations, where 2sin(θ) = -1.
- The equation 5tan(θ) = 4 leads to solutions approximately at 38.66° and 218.66°.
- For 5cos(θ) = -3, solutions include 126.87°, 53.13°, and 233.13°.
- The equation 2cos²(θ) = 1 gives angles including 45°, 135°, 225°, and 315°.
Geometry Applications
- In a geometric scenario, bc = 1 and ef = 3, with ac = 2, leads to finding df corresponding to variable relationships.
- For jk = 4 and mn = 10 with jl = 8, the task is to determine mo.
Calculations
- The equation c² = 4² + 6² evaluates to √(52), providing a distance interpretation.
- The calculation 8 × 10^6 × log(10) simplifies to 69, showcasing exponential growth or scientific notation use.
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Description
This quiz covers fundamental concepts in trigonometric and logarithmic functions, including key values, identities, and transformations. Explore angles, function behaviors, and how logarithms illustrate exponentiation through various properties. Perfect for understanding the interplay between these essential mathematical topics.
