Angles and Their Types

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Questions and Answers

What is the correct definition of the initial side of an angle?

The position of the ray after rotation.

The common endpoint shared by the two rays.

The angle formed between two intersecting lines.

The initial position of the ray before rotation. (correct)

Which of the following describes a standard angle?

An angle with the initial side on the y-axis.

An angle with its initial side coinciding with the x-axis and vertex at the origin. (correct)

An angle that measures less than 90 degrees.

An angle with its vertex at any point on a circle.

What is the measure of one complete revolution in radians?

π radians

360°

180°

2π radians (correct)

Identify the type of angle that is greater than 0 degrees but less than 90 degrees.

Acute angle Signup and view all the answers

What is the measure of an angle if it represents a quarter of a complete revolution in degrees?

90° Signup and view all the answers

What is the coterminal angle of $rac{7 heta}{3}$?

$rac{7 heta}{3} - 2 heta$ Signup and view all the answers

Which of the following is a reference angle for an angle of 151°?

29° Signup and view all the answers

Which trigonometric function is equivalent to the ratio $rac{y}{r}$?

sine Signup and view all the answers

In which quadrants is the cosine function positive?

I and IV Signup and view all the answers

What is the reference angle of -269°?

91° Signup and view all the answers

What is the length of the hypotenuse for the point P(-5, 12)?

$ ext{13}$ Signup and view all the answers

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

Sine Signup and view all the answers

What is the value of the cosine of angle θ for the point P(-5, 12)?

$\frac{-5}{13}$ Signup and view all the answers

What is the cotangent of angle θ for the point P(-5, 12)?

$\frac{5}{-12}$ Signup and view all the answers

Which of the following statements is correct regarding the trigonometric functions for angle θ?

Cosecant is the reciprocal of sine. Signup and view all the answers

Which equation represents a contradiction equation?

x² = x² + 3 Signup and view all the answers

What is the reciprocal identity of tan θ?

cot θ Signup and view all the answers

Which of the following is a conditional equation?

x + 2 = 5 Signup and view all the answers

Which Pythagorean identity correctly represents the relationship between tangent and secant?

1 + tan² θ = sec² θ Signup and view all the answers

Which of the following pairs represents equivalent expressions according to quotient identities?

tan θ and sin θ/cos θ Signup and view all the answers

Study Notes

Angles

An angle is formed by two rays or lines that share a common endpoint.
The rays are called the sides of the angle.
The initial side is the starting position of the ray.
The terminal side is the position of the ray after rotation.
A standard angle has its vertex at the origin and its initial side coincides with the x-axis.
The measure of an angle is the amount of rotation needed to move from the initial side to the terminal side.
Angles can be measured in degrees or radians:
- 360° = 2π radians (one full circle)
- π radians = 180°
Types of angles:
- Acute angles are between 0° and 90°.
- Right angles are equal to 90°.
- Obtuse angles are between 90° and 180°.
- Straight angles are equal to 180°.
- Reflex angles are greater than 180° and less than 360°.
Coterminal angles share the same initial and terminal sides.
A reference angle is the acute angle formed between the terminal side and the x-axis.
To convert from radians to degrees, multiply by $\frac{180}{\pi}$.
To convert from degrees to radians, multiply by $\frac{\pi}{180}$.

Circular Functions

Trigonometric functions relate the angles of a right triangle to the ratios of its sides.
SOH CAH TOA is a mnemonic to remember the trigonometric functions:
- Sine = Opposite / Hypotenuse
- Cosine = Adjacent / Hypotenuse
- Tangent = Opposite / Adjacent
The reciprocal trigonometric functions are:
- Cosecant (csc θ) = 1 / sin θ
- Secant (sec θ) = 1 / cos θ
- Cotangent (cot θ) = 1 / tan θ
Pythagorean Identities: Fundamental relationships between trigonometric functions:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = csc² θ
Quotient Identities: Express tangent and cotangent in terms of sine and cosine:
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ

Evaluating Trigonometric Functions

To evaluate trigonometric functions when given a point on the terminal side of an angle:
- Determine the value of r (the hypotenuse) using the Pythagorean theorem: r² = x² + y²
- Apply the SOH CAH TOA definitions to find the values of the trigonometric functions.
For the point P(-5, 12):
- r = √((-5)² + (12)²) = 13
- sin θ = y/r = 12/13
- cos θ = x/r = -5/13
- tan θ = y/x = -12/5
- csc θ = r/y = 13/12
- sec θ = r/x = -13/5
- cot θ = x/y = -5/12

Trigonometric Identities

An identity equation is true for all possible values of the variable (domain).
A conditional equation is true for specific values of the variable.
A contradiction equation is never true for any value of the variable.
Fundamental Trigonometric Identities:
- Reciprocal Identities: connect a function with its reciprocal, like sin θ and csc θ.
- Quotient Identities: show how to express tangent and cotangent in terms of sine and cosine.
- Pythagorean Identities: show a relationship between the squares of sine and cosine functions.

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Description

This quiz covers the fundamental concepts of angles, including their definitions, types, and measurements in both degrees and radians. You'll learn about acute, right, obtuse, straight, and reflex angles, as well as related concepts like coterminal and reference angles. Test your understanding of this essential topic in geometry!