Trigonometry Chapter Review

Questions and Answers

What are the co-terminal angles for an angle?

±360º

±2π

±360º or ±2π (correct)

None of the above

How many radians are in one full revolution?

6.28 radians (or 2π radians)

To convert from degrees to radians, you multiply by?

π / 180º

To convert from radians to degrees, you multiply by?

180º / π Signup and view all the answers

The formula for Arc Length in degrees is?

(θº/360º) ∙ 2πr Signup and view all the answers

The formula for Arc Length in radians is?

rθ Signup and view all the answers

In which quadrant are all trigonometric ratios positive?

Quadrant 1 Signup and view all the answers

In which quadrant is only sine positive?

Quadrant 2 Signup and view all the answers

In which quadrant is only tangent positive?

Quadrant 3 Signup and view all the answers

In which quadrant is only cosine positive?

Quadrant 4 Signup and view all the answers

What is the formula for Sec?

1/cosθ Signup and view all the answers

What is the formula for Csc?

1/sinθ Signup and view all the answers

What is the formula for Cot?

1/tanθ Signup and view all the answers

How is tanθ expressed in terms of sine and cosine?

(sinθ/cosθ) Signup and view all the answers

What is the Pythagorean identity?

sin²θ + cos²θ = 1 Signup and view all the answers

What does Amplitude equal in the function y= a sin (bx + c) + d?

|a| Signup and view all the answers

What is the general equation for sine graphs?

y= a sin (bx + c) + d Signup and view all the answers

What is the formula for Period?

2π / b Signup and view all the answers

What is Gap in relation to the period?

period / 4 Signup and view all the answers

What does Vertical Shift equal in oscillation?

d Signup and view all the answers

What is the formula for Phase Shift?

c / b Signup and view all the answers

What is the formula used to calculate the start of the wave?

c / b (or phase shift) Signup and view all the answers

What is the formula for the end of the wave?

start + period Signup and view all the answers

How is the top of the wave determined?

Vertical shift + amplitude Signup and view all the answers

What represents the middle of the wave?

vertical shift Signup and view all the answers

What represents the bottom of the wave?

vertical shift - amplitude Signup and view all the answers

What is the period of tangent and cotangent?

π / b Signup and view all the answers

What are the asymptotes of tangent?

(± period / 2) ± per Signup and view all the answers

What are the asymptotes of cotangent?

0 ± π Signup and view all the answers

Study Notes

Co-terminal Angles

Co-terminal angles can be determined by adding or subtracting multiples of ±360º or ±2π.

Radians in a Full Revolution

One full revolution equals 6.28 radians, or equivalently, 2π radians.

Conversion Formulas

To convert degrees to radians, multiply degrees by π / 180º.
To convert radians to degrees, multiply radians by 180º / π.

Arc Length Formulas

When measuring in degrees, arc length is calculated as (θº/360º) ∙ 2πr, where θ is in degrees and r is the radius.
In radians, arc length is found using the formula arc length = rθ, where θ is in radians.

Reference Angle

The reference angle is the acute angle created between the terminal side of an angle and the x-axis; it is always positive.

Quadrants and Trigonometric Ratios

In Quadrant I, all trigonometric ratios are positive.
In Quadrant II, only the sine function is positive.
In Quadrant III, only the tangent function is positive, as sine and cosine are both negative.
In Quadrant IV, only the cosine function is positive.

Key Trigonometric Functions

Secant: ( \sec \theta = \frac{1}{\cos \theta} )
Cosecant: ( \csc \theta = \frac{1}{\sin \theta} )
Cotangent: ( \cot \theta = \frac{1}{\tan \theta} )
Tangent: ( \tan \theta = \frac{\sin \theta}{\cos \theta} )
Cotangent: ( \cot \theta = \frac{\cos \theta}{\sin \theta} )

Pythagorean Identity

The fundamental Pythagorean identity states that ( \sin^2 \theta + \cos^2 \theta = 1 ).

Wave Properties

Amplitude of a sine function is represented as |a| in the equation ( y = a \sin(bx + c) + d ).
General equation for sine graphs is ( y = a \sin(bx + c) + d ).

Period and Shifts

The period of a sine or cosine function is determined by ( \text{Period} = \frac{2π}{b} ).
Gap, or interval between key points, is defined as ( \text{period} / 4 ).
Vertical shift is represented by the value d in the function.
Phase shift is calculated using ( \text{phase shift} = \frac{c}{b} ).

Graphing Sine Functions

Starting point of the graph is positioned at the phase shift ( c/b ).
The endpoint can be determined by adding the period to the starting point.
The top of the wave is at the vertical shift plus amplitude, while the bottom is at the vertical shift minus amplitude.

Tangent and Cotangent Functions

The period of tangent and cotangent functions is given by ( \frac{π}{b} ).
Asymptotes for the tangent function occur at ( (±\text{period}/2) ± \text{period} ).
Asymptotes for the cotangent function can be found at ( 0 ± π ).

Studying That Suits You

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Description

Test your knowledge on co-terminal angles, radians, and arc length formulas with this quiz. You'll also explore the relationship between degrees and radians and understand trigonometric ratios across different quadrants. Perfect for reinforcing key concepts in trigonometry!