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Trigonometry Basics Quiz

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Questions and Answers

What is the ratio represented by the cosine function?

Hypotenuse to adjacent side

Opposite side to adjacent side

Opposite side to hypotenuse

Adjacent side to hypotenuse (correct)

If sin(A) = 4/5, what does this imply about angle A in a right triangle?

A is 90 degrees

A is 37 degrees

A is 60 degrees

A is 53.13 degrees (correct)

What is the value of tan(45 degrees) in a right triangle?

Undefined

-1

1 (correct)

When simplifying trigonometric expressions, what is the purpose of moving fractions into a single denominator?

To facilitate further calculations by making operations easier Signup and view all the answers

Which trigonometric function represents the ratio of the opposite side to the hypotenuse?

Sine (sin) Signup and view all the answers

What does cos(0 radians) equal to?

1 Signup and view all the answers

What is the simplified form of sin^(-1)(-sqrt(3)/2) + 1/(2sqrt(2)pi) + cos(pi/4) / (sqrt(2)*sqrt(pi))?

sin^(-1)(-sqrt(3)/2) + pi/2 + cos(pi/4) * (1/sqrt(2)*sqrt(pi)) Signup and view all the answers

What is the value of x that satisfies the equation sin(x) = -1 / sqrt(2)?

-pi/4 Signup and view all the answers

Which trigonometric identity is always true for all values of the input?

sin^2(θ) + cos^2(θ) = 1 Signup and view all the answers

What is the result of combining the first two terms in the expression sin^(-1)(-sqrt(3)/2) + 1/(2sqrt(2)pi)?

-sin^(-1)(-sqrt(3)/2) - pi/2 Signup and view all the answers

Which term should be added to cos(pi/4) to simplify the expression sin^(-1)(-sqrt(3)/2) + pi/2?

(cos(pi/4)) * (1/sqrt(2)*sqrt(pi)) Signup and view all the answers

What is the correct form of the Pythagorean identity?

<code>a^2 + b^2 = c^2</code> Signup and view all the answers

Study Notes

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).

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Description

Test your knowledge of trigonometry basics including trigonometric functions, simplifying trigonometric expressions, solving trigonometric equations, and common trigonometric identities. This quiz covers fundamental concepts and techniques in trigonometry.