Practice Questions

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

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

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

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

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

Name the six trigonometric functions and list their reciprocal pairs.

Recall the formula for tan(x - y).

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.

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

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

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

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.

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

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

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

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 $\pi = \frac{22}{7}$)

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

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

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.

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).

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).

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).

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.

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.

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

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

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

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).

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 $\cos 20^\circ \cos 40^\circ \cos 80^\circ$.

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

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