Trigonometric Identities and Equations Quiz

Different trigonometric expressions that represent the same graph.

Shifting a trigonometric graph left or right.

A trigonometric function repeats itself at regular intervals.

A function that is symmetrical about the y-axis (f(x) = f(-x)).

A function that is symmetrical about the origin (f(-x) = -f(x)).

Two angles that add up to 90 degrees.

An acute angle formed by the given angle's reference point and the nearest horizontal axis in the unit circle

Shifting cosine function right π/2 produces sine function.

The sine of an angle θ is equal to the cosine of its complementary angle (π/2 - θ).

Cos(π/3) is equal to 1/2.

Tan(π/6) is equal to 1/√3.

Tan(θ) is equal to Cot(π/2 - θ).

Given acute angle θ in Quadrant I, the related acute angle in Quadrant II is θ (the same as the original angle) but used with the appropriate signs for that quadrant.

Given acute θ in Quadrant I, the related acute angle in Quadrant III is θ, with both sine and cosine functions being negative.

The related acute angle in Quadrant IV is θ, used with the appropriate signs for that quadrant.

Trigonometric identities that express one function in terms of the reciprocal of another function. For example, csc θ = 1/sin θ.

Trigonometric identities that express one function in terms of the ratio of two other functions. For example, tan θ = sin θ / cos θ.

Trigonometric identities that relate the squares of sine, cosine, tangent, cotangent, secant, and cosecant. For example, sin²θ + cos²θ = 1.

Trigonometric identities that express the sine, cosine, or tangent of twice an angle in terms of trigonometric functions of the original angle. For example, sin 2θ = 2 sin θ cos θ.

Trigonometric identities that express the sine, cosine, or tangent of the sum or difference of two angles in terms of trigonometric functions of the individual angles. For example, sin(a + b) = sin a cos b + cos a sin b.

An angle formed by adding or subtracting two or more angles.

Precise values of trigonometric functions, often involving radicals, obtained through compound angle formulas.

sin(a + b) = sin a cos b + cos a sin b

cos(a - b) = cos a cos b + sin a sin b

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²θ - sin²θ

An equation involving trigonometric functions that is always true for all permissible values of the variable.

Demonstrating that two trigonometric expressions are equivalent using algebraic manipulations and trigonometric identities.

Solve by factoring like a quadratic equation, using a substitution for the trig function. Then, solve for the trig function and find the required angles.

If factoring is difficult, use the quadratic formula to solve for the trig function. Then, find the angles that satisfy the results.

CAST is used to determine all solutions to equations within a given interval, using the quadrant signs of the trig function (sin, cos, tan).

First, find the related acute angle. Then, use CAST to determine the quadrants where the trig function is positive or negative and solve for the angle in each quadrant.

To find ALL solutions, add the period of the trig function to each angle obtained from CAST.

Solve 2sin(x)cos(x) = cos(2x) for 0 ≤ x ≤ 2π. First, rewrite using the double-angle formula, then use CAST to find all solutions.

Use trigonometric functions to model real-world phenomena that exhibit periodic behavior. For example, using a cosine function to model tidal patterns.

To find specific times when a modeled condition is met, solve the trigonometric equation for the variable representing time. Use the model's equation and CAST to find all solutions.

This identity states that the cosine of an angle that is π/2 greater than another angle is equal to the negative of the sine of that other angle.

This identity expresses the ratio of two cosine expressions, one with the difference of two angles and the other with their sum, in terms of tangent functions.

This identity simplifies the expression on the left side to the double angle sine of the angle x.

Trigonometric equations involve trigonometric functions and have solutions that occur at regular intervals because of the periodic nature of these functions.

The sine function has 6 solutions within this interval. It takes on values of 0, 1, -1 and has 2 solutions for values between 0 and 1 and between -1 and 0.

The cosine function has 8 solutions in the given interval. It takes on values of 0, 1, -1 and has 4 solutions for values between 0 and 1 and between -1 and 0.

To solve a trigonometric equation, isolate the trigonometric function, determine the quadrant(s) where the function has the desired value, find the related acute angle, and then use the CAST rule and function's period to determine all solutions within the specified interval.

The solutions for this equation are found within the specified interval by considering that cosine is negative in the second and third quadrants, finding the related acute angle, and applying the CAST rule and the function's period.