Trigonometry Basics Quiz

Introduction to Trigonometry

Trigonometry is a branch of mathematics that deals with relationships between angles and side lengths of triangles. It's particularly useful when dealing with right triangles—those where one angle measures 90 degrees, formed by two perpendicular sides called the legs (the shorter sides) and hypotenuse (the longer side opposite the right angle).

Trigonometric Functions

The three most common trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each function represents a ratio that describes a relationship within a triangle:

• Sine (sin): the ratio of the length of the side opposite the given angle to the length of the longest side of the triangle; sin(A)=O/H, where O is the length of the side opposite the angle A.
• Cosine (cos): the ratio of the length of the adjacent side to the length of the longest side; cos(A)=A/H.
• Tangent (tan): the ratio of the length of the side opposite the given angle to the length of the adjacent side; tan(A) = O/A.

These functions can also be calculated using different formulas based on radian measure instead of degree measure.

Simplifying Trigonometric Expressions

Simplification often involves moving fractions into a single denominator, which can be done by multiplying both numerator and denominator with a fraction that has the desired term as its numerator. For example, simplify:

sin^(-1)(-sqrt(3)/2) + 1/(2*sqrt(2)*pi) + cos(pi/4) / (sqrt(2)*sqrt(pi))

First combine the first two terms using +:

sin^(-1)(-sqrt(3)/2) + 1/(2*sqrt(2)*pi) + cos(pi/4) / (sqrt(2)*sqrt(pi))

Then simplify the second term using sqrt():

sin^(-1)(-sqrt(3)/2) + pi/2 + cos(pi/4) / (sqrt(2)*sqrt(pi))

Finally, combine the last two terms using +:

sin^(-1)(-sqrt(3)/2) + pi/2 + cos(pi/4) * (1/sqrt(2)*sqrt(pi))

This expression is now in its simplest form.

Solving Trigonometric Equations

Solving trigonometric equations usually involves determining values of x that make the equation true. This can be done by finding the inverse of each trigonometric function involved in the equation:

Example:

sin(x) = -1 / sqrt(2)

Inverse sine of (-1 / sqrt(2)):

x = -pi/4

So x=-π/4 satisfies the equation.

Trigonometric Identities

Trigonometric identities are mathematical statements that are always true for all values of the input. Some common identities include:

• Pythagorean identity: a^2 + b^2 = c^2, where c is the hypotenuse of a right triangle with legs a and b.
• Other basic identities: sin^2(θ) + cos^2(θ) = 1 and csc(α)cot(α) = sec(α).
• Product-to-sum formula: sin(x + y) = sin(x)cos(y) + cos(x)sin(y).