Trigonometry and Logarithmic Functions Quiz

The argument is the input value that results in a specific output for a logarithmic function. In this case, the argument of the logarithmic function 'log₃ 9 = 2' is the number 9, because when we input 9 into the logarithm, we get an output of 2. Therefore, the argument of the logarithmic function is the value that we are taking the logarithm of, which is 9. The argument refers to the value inside the logarithmic function, which is 9 in this case.

The value of cos 30° is expressed as √3/2, which is approximately 0.86602540378. It can be derived using the properties of right-angled triangles. A right-angled triangle with an angle of 30° has a hypotenuse twice the length of its shorter side. The cosine of an angle is determined by the ratio of the adjacent side to the hypotenuse, which is √3/2 in this case.

The reciprocal function of sin θ is csc θ. In trigonometry, reciprocals of functions play a crucial role in simplifying equations and expressing relationships between trigonometric ratios. The reciprocal of a function is simply its multiplicative inverse, meaning that when multiplied together, they result in 1. Therefore, the reciprocal of sin θ is csc θ, and vice versa.

The quadrant where tan θ > 0 and sin θ < 0 is quadrant III. This determination arises from the understanding of the signs of trigonometric functions in different quadrants. In Quadrant III, both the x-coordinate and the y-coordinate are negative. Since tan θ = sin θ/cos θ, and both sin θ and cos θ are negative in Quadrant III, their ratio, tan θ, becomes positive. On the other hand, sin θ is negative in Quadrant III because the y-coordinate is negative.

90 degrees is equal to π/2 radians. This conversion is based on the fundamental relationship between degrees and radians – a complete circle is 360 degrees, which is equal to 2π radians. Therefore, 90 degrees, being a quarter of a full circle, corresponds to π/2 radians.

Shifting a graph upwards or downwards involves adding or subtracting a constant value to the entire function. In this instance, shifting the graph of y = x + 4 to y = x - 4 requires subtracting 8 units from the original function, which results in moving every point on the graph 8 units downwards. The new equation, y = x - 4, represents the same linear function as y = x + 4, but shifted downwards by 8 units.

A one-to-one function is a function where each input value has a unique output value. However, the function y = x² + 8 is not a one-to-one function because multiple input values can lead to the same output value. For instance, both x = 2 and x = -2 would produce the same output of y = 12. Therefore, the function is not one-to-one as it does not fulfill the condition of a unique output for each input.

The equation 2 = log₆ 36 is true because it correctly expresses the relationship between the base 6, the exponent 2, and the result 36. This equation states that 6 raised to the power of 2 equals 36, which is true according to the definition of logarithms. In general, the equation logₐ b = c implies that a raised to the power of c equals b, meaning that the base raised to the power of the logarithm's result equals the argument.

The cotangent of an angle (cot θ) can be expressed in terms of the adjacent (adj) and opposite (opp) sides of a right-angled triangle. In a right-angled triangle, cot θ is defined as the ratio of the adjacent side to the opposite side. This is the reciprocal of the tangent function (tan θ), which is defined as the ratio of the opposite side to the adjacent side. Therefore, cot θ can be equivalently written as adj/opp.

The reference angle for 145° is 35°. Reference angles are acute angles formed by the terminal side of an angle and the x-axis. They play a crucial role in simplifying trigonometric calculations. To find the reference angle, subtract the angle from 180° or 360°, depending on where the angle lies. In this case, since 145° is in Quadrant II, we subtract it from 180°, resulting in a reference angle of 35°.

The expression for log(√x/y) in terms of a single logarithm can be simplified by using the properties of logarithms. The properties of logarithms state that the logarithm of a quotient is equivalent to the difference of the logarithms of the numerator and denominator, and the logarithm of a power is equivalent to the product of the power and the logarithm of the base. Therefore, log(√x/y) can be expressed as (1/2) log x - log y.

The expression log x + 2 log y can be simplified to log(xy²) by using the properties of logarithms. The logarithmic property states that the logarithm of a product is equivalent to the sum of the logarithms of the factors. Additionally, the logarithm of a power is equivalent to the product of the power and the logarithm of the base. Applying these properties, we can combine log x + 2 log y into a single logarithm, yielding log(xy²)

The value of x in the equation 5^x = 625 is 4 because 5 raised to the power of 4 equals 625. This equation is a simple exponential equation where we need to find the exponent that will make the base (5) equal to the result (625). In general, if a^x = b, then x = logₐ b. To solve this equation, we can take the logarithm of both sides with base 5 to find the value of x, which is 4.

The values of x in the logarithmic equation log x² + log 2 = log 8 are ±2. This equation involves logarithmic properties that can be utilized to solve for x. Applying these properties, we simplify the equation to log (2x²) = log 8. Then, we can conclude that 2x² = 8. Therefore, the possible values of x are ±2.

The tangent of 45 degrees (tan 45°) is equal to 1. This value can be understood using the concept of a right-angled triangle. In a right-angled triangle with an angle of 45 degrees, the opposite and adjacent sides are equal. Using the definition of tangent (tan θ = opposite/adjacent), we can see that tan 45° = 1, since the opposite and adjacent sides are equal.

The sine of 330 degrees (sin 330°) is equal to -1/2. This value can be understood using the unit circle and the concept of reference angles. The reference angle for 330 degrees is 30 degrees, and since 330 degrees lies in the fourth quadrant, where sine is negative, sin 330° is equal to -sin 30° which is -1/2.

To solve for θ in the equation 5 tan θ = 4, we need to isolate tan θ by dividing both sides by 5, resulting in tan θ = 4/5. Then, we use the arctangent function (arctan) to find the value of θ. The arctangent function provides the angle whose tangent is a given value. In this case, arctan(4/5) ≈ 38.66 degrees. However, the tangent function has a period of 180 degrees, meaning that it repeats every 180 degrees. Therefore, the general solution for θ is 38.66° + 180°n, where 'n' is an integer.

To solve for θ in the equation 2 sin θ = -1, we first isolate sin θ by dividing both sides by 2, resulting in sin θ = -1/2. Then, we need to find the angles θ where sin θ = -1/2. The sine function is negative in both the third and fourth quadrants. In the fourth quadrant, the reference angle is 30 degrees, and in the third quadrant, the reference angle is 150 degrees. Therefore, the possible solutions for θ are 210 degrees (180 degrees + 30 degrees) and 330 degrees (360 degrees - 30 degrees) and additional solutions can be found by adding multiples of 360 degrees.

To solve for θ in the equation 5 cos θ = -3, we first isolate cos θ by dividing both sides by 5, resulting in cos θ = -3/5. Then, we need to find the angles θ where cos θ = -3/5. The cosine function is negative in both the second and third quadrants. In the second quadrant, the reference angle is approximately 53.13 degrees, and in the third quadrant, it's also 53.13 degrees. Therefore, the possible solutions for θ are 126.87 degrees (180 degrees - 53.13 degrees) and 233.13 degrees (180 degrees + 53.13 degrees) and additional solutions can be found by adding multiples of 360 degrees.

To solve for θ in the equation 2 cos² θ= 1, we first simplify the equation by dividing both sides by 2, resulting in cos² θ = 1/2. Then, we take the square root of both sides, resulting in cos θ = ±√(1/2) = ± √2 / 2. We then find the angles where cos θ = √2 / 2 and cos θ = -√2 / 2. The cosine function is positive in the first and fourth quadrants, and negative in the second and third quadrants. In the first quadrant, the reference angle for cos θ = √2 / 2 is 45 degrees, and in the fourth quadrant, the reference angle is also 45 degrees. In the second quadrant, the reference angle for cos θ = -√2 / 2 is 45 degrees, and in the third quadrant, the reference angle is also 45 degrees. Therefore, the possible solutions for θ are 45 degrees, 135 degrees, 225 degrees, and 315 degrees and additional solutions can be found by adding multiples of 360 degrees.

df is 6. In similar figures, corresponding sides are proportional. Since bc corresponds to ef and ac corresponds to df, we can write the proportion bc/ef = ac/df. Substituting the given values, we have 1/3 = 2/df. Solving for df, we get df = 6.

mo is 20. Similar figures have corresponding sides that are proportional. Since jk corresponds to mn and jl corresponds to mo, we can set up the proportion jk/mn = jl/mo. Substituting the given values, we have 4/10 = 8/mo. Solving for mo, we get mo = 20.

The cosine of 225 degrees (cos 225°) is -√2/2. This value can be determined using the unit circle and the concept of reference angles. Since 225 degrees lies in Quadrant III, where cosine is negative, and its reference angle is 45 degrees, we can conclude that cos 225° is -√2/2.

The sine of 225 degrees (sin 225°) is -√2/2. This value can be determined using the unit circle and the concept of reference angles. Since 225 degrees lies in Quadrant III, where sine is negative, and its reference angle is 45 degrees, we can conclude that sin 225° is -√2/2.

The tangent of 225 degrees (tan 225°) is 1. This value can be determined using the concept of trigonometric ratios. Since 225 degrees lies in Quadrant III, where both sine and cosine are negative, and their magnitudes are equal, the tangent of 225 degrees will be 1, as the negatives cancel out.

The value of c is 7.2. This is based on the Pythagorean Theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides. Therefore, c² = 4² + 6² = 52. Taking the square root of both sides gives us c = √52, which is approximately 7.2.

The expression 8 * 10⁶ * log 10 equals 69. This is derived by using the properties of logarithms and the understanding that log 10 = 1. Logarithms are used to express power relationships. In this case, log 10 represents the power to which 10 must be raised to obtain 10, which is 1. Therefore, 8 * 10⁶ * log 10 is the same as 8 * 10⁶ * 1, resulting in 8 * 10⁶ which is equivalent to 69.