Trigonometry and Logarithmic Functions Quiz

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Questions and Answers

What is the value of cos 30°?

$\frac{\sqrt{3}}{2}$

What is the argument of $2 = \log_3 9$?

What is the reciprocal function of sin θ?

csc θ

In which quadrant is tan > 0 and sin < 0?

III (B) Signup and view all the answers

What is 90° in radians?

$\frac{\pi}{2}$ Signup and view all the answers

What is the equation y = x + 4 shifted to y = x - 4?

y = x - 4 Signup and view all the answers

Is y = x^2 + 8 a one-to-one function?

False (B) Signup and view all the answers

Is $6^2 = 36$ correct according to the equation $2 = \log_6 36$?

True (A) Signup and view all the answers

What is the value of cot θ in terms of adj/opp?

$\frac{adj}{opp}$ Signup and view all the answers

What is the reference angle for 145°?

35° Signup and view all the answers

What is the expression for $\log \left( \frac{\sqrt{x}}{y} \right)$ in terms of a single logarithm?

$\frac{1}{2} \log x - \log y$ Signup and view all the answers

What is the expression for $\log x + 2 \log y$?

$\log(xy^2)$ Signup and view all the answers

What is x in the equation $5^x = 625$?

4 Signup and view all the answers

What are the values for $\log x^2 + \log 2 = \log 8$?

$\pm 2$ Signup and view all the answers

What is the value of tan 45°?

1 Signup and view all the answers

What is the value of sin 330°?

$-\frac{1}{2}$ Signup and view all the answers

Solve for θ: $5 \tan θ = 4$.

38.66; 218.66 Signup and view all the answers

Solve for θ: $2 \sin θ = -1$.

30; 330; 210 Signup and view all the answers

Solve for θ: $5 \cos θ = -3$.

126.87; 53.13; 233.13 Signup and view all the answers

Solve for θ: $2 \cos^2 θ = 1$.

45; 135; 225; 315 Signup and view all the answers

If bc = 1 and ef = 3 when ac = 2, what is df?

6 Signup and view all the answers

If jk = 4 and mn = 10 when jl = 8, what is mo?

20 Signup and view all the answers

What is the value of cos 225°?

$-\frac{\sqrt{2}}{2}$ Signup and view all the answers

What is the value of sin 225°?

$-\frac{\sqrt{2}}{2}$ Signup and view all the answers

What is the value of tan 225°?

1 Signup and view all the answers

Solve for c: $c^2 = 4^2 + 6^2 = \sqrt{52}$.

7.2 Signup and view all the answers

What does $8 \times 10^6 \times \log 10$ equal?

69 Signup and view all the answers

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.