Practice Questions

Define the term radian measure as it relates to an angle in a circle.

State the formula that expresses the relationship between degree measure and radian measure.

Calculate the value of $\cos(-750^\circ)$.

A pendulum of length 50 cm swings through an angle. If its tip describes an arc of length 10 cm, calculate the angle of the swing in radians.

Recall the formula for tan(x - y).

Calculate the value of $\sin 75^\circ + \sin 15^\circ$.

Convert the angle $\frac{11\pi}{12}$ radians into degree measure.

Name the six trigonometric functions and list their reciprocal pairs.

Calculate the degree measure of an angle that is 5π/12 radians.

Examine the following statement and state if it is true or false: The equation sin(x) = 2 has a real solution for x.

State the value of sin(π) and cos(π/2).

Two wheels are rotated. The first turns through 6 radians in a second, while the second turns through 450 revolutions per minute. Compare their angular speeds and determine which wheel is rotating faster.

Explain the concept of quadrantal angles and provide two examples in degrees.

Recall the formula for sin(2x) in terms of both sine/cosine and in terms of tangent.

If sin(x) = -4/5 and x lies in the third quadrant, solve for the values of tan(x) and sec(x).

Arcs of the same length in two circles subtend angles of $120^\circ$ and $75^\circ$ at their centers. Analyze the relationship to find the ratio of their radii.

If $\sec x = -\frac{13}{12}$ and $\frac{\pi}{2} < x < \pi$, calculate the value of the expression $2 \sin x + 3 \cos x$.

Calculate the radius of a circle where a central angle of 60° intercepts an arc of length 44 cm. Use π = 22/7.

If $\tan x = -\frac{4}{3}$ and $x$ lies in the second quadrant, analyze the signs to find the value of $\sin x$.

Calculate the length of an arc of a circle with a radius of 14 cm that subtends an angle of $45^\circ$ at the center. (Use $\pi = \frac{22}{7}$)

Convert $-22^\circ 30'$ into radian measure.

If $\cos x = -\frac{3}{5}$ and $x$ lies in the third quadrant ($\pi < x < \frac{3\pi}{2}$), analyze the quadrant of $\frac{x}{2}$ and calculate the values of $\sin(\frac{x}{2})$, $\cos(\frac{x}{2})$, and $\tan(\frac{x}{2})$.

Demonstrate that $(\cos x + \cos y)^2 + (\sin x + \sin y)^2 = 4 \cos^2\left(\frac{x-y}{2}\right)$.

Demonstrate the calculation of the exact value of tan(105°) by applying the sum identity for tangent.

Calculate the value of cosec(-1110°).

Given tan(θ) = 3/4 and θ lies in the third quadrant, calculate the value of cos(2θ).

Apply a suitable identity to demonstrate that the expression sin(75°) - sin(15°) is equal to cos(45°).

Analyze the expression tan(x) + cot(x) and demonstrate that it can be simplified to 2cosec(2x).

If $\sin A = \frac{4}{5}$ and $A$ is in the second quadrant, calculate the value of $\sin 2A$.

The hour hand of a clock is 6 cm long. Calculate the distance its tip moves in 20 minutes. (Use $\pi = 3.14$)

List the signs of all six trigonometric functions when an angle's terminal side lies in the third quadrant.

Explain the convention used to determine if an angle is positive or negative.

Summarize the relationship between the length of a circular arc, the radius of the circle, and the central angle subtended by the arc.

Identify the intervals between 0 and 2π where the cosine function is positive and where it is negative.

Recall the sum-to-product formula for cos x + cos y.

Describe the domain and range of the standard tangent function, y = tan x.

List the four different formulas for cos(2x) given in the chapter.

Solve the equation 2cos^2(x) + 3sin(x) - 3 = 0 for principal values of x.

If sin(A) = 5/13 and cos(B) = 4/5, where A is in the second quadrant and B is in the fourth quadrant, calculate the value of sin(A - B).

State the fundamental Pythagorean identity for trigonometry and explain how the other two Pythagorean identities are derived from it.

Demonstrate the proof of the trigonometric identity: (cos(4x) + cos(3x) + cos(2x)) / (sin(4x) + sin(3x) + sin(2x)) = cot(3x).

Calculate the value of the product $\sin 10^\circ \sin 30^\circ \sin 50^\circ \sin 70^\circ$.

If tan(x) = -4/3 and x lies in the second quadrant, solve for the values of sin(x/2) and cos(x/2).

Calculate the value of the expression $\cos 20^\circ \cos 40^\circ \cos 80^\circ$.

Calculate the value of $\tan 105^\circ$ by expressing it as a sum of two standard angles.