Basic Calculus Module 4: Trigonometry

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Questions and Answers

What is the Pythagorean Identity expressed in terms of sine and cosine?

$ ext{sin}^2 heta - ext{tan}^2 heta = 1$

$ ext{sin}^2 heta + ext{cos}^2 heta = 1$ (correct)

$ ext{sin}^2 heta - ext{cos}^2 heta = 0$

$ ext{sin}^2 heta + ext{cos}^2 heta = ext{tan}^2 heta$

In which quadrant are both sine and cosine positive?

Quadrant I (correct)

Quadrant II

Quadrant IV

Quadrant III

What is the value of sin(60°)?

$rac{√3}{2}$ (correct)

$rac{√2}{2}$

$rac{1}{2}$

$1$

How is the Pythagorean Identity used in trigonometry?

To simplify expressions containing sine and cosine. Signup and view all the answers

Which of the following represents a common Pythagorean identity?

$sec^2(\theta) - tan^2(\theta) = 1$ Signup and view all the answers

Given that $sin(\theta) \neq 0$, which identity is valid?

$cot^2(\theta) = csc^2(\theta) - 1$ Signup and view all the answers

When verifying the identity $(sec(\theta) - tan(\theta))(sec(\theta) + tan(\theta)) = 1$, which step involves an identity?

Expand to $sec^2(\theta) - tan^2(\theta)$ Signup and view all the answers

What transformation verifies the identity $tan(\theta) = sin(\theta) sec(\theta)$?

Setting $sec(\theta) = \frac{1}{cos(\theta)}$ Signup and view all the answers

Which of the following identities is not a standard Pythagorean identity?

$tan^2(\theta) + sec^2(\theta) = 1$ Signup and view all the answers

If $sec(\theta) = \frac{1}{cos(\theta)}$, how can you express $tan(\theta)$?

$tan(\theta) = sin(\theta) sec(\theta)$ Signup and view all the answers

How would you express the relationship $5sec(\theta)tan(\theta)$ using Pythagorean identities?

$5sec^2(\theta)$ Signup and view all the answers

What is the correct identity involving $csc(\theta)$ and $cot(\theta)$?

$csc^2(\theta) = 1 + cot^2(\theta)$ Signup and view all the answers

Which of the following represents a common alternate form of the Pythagorean identity?

𝑐𝑜𝑠2 (𝜃) = 1 − 𝑠𝑖𝑛2 (𝜃) Signup and view all the answers

What is the identity derived from dividing the Pythagorean identity by cosine squared?

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

Which Pythagorean identity requires that sin(𝜃) ≠ 0 for its derivation?

𝑐𝑜𝑡2 (𝜃) + 1 = 𝑐𝑠𝑐2 (𝜃) Signup and view all the answers

Which of the following is NOT a Pythagorean identity?

𝑐𝑜𝑠2 (𝜃) + 𝑐𝑜𝑡2 (𝜃) = 1 Signup and view all the answers

What is the correct form for the Quadrant II solution given in the content?

𝜃 = rac{3𝜋}{4} + 2𝜋𝑘 Signup and view all the answers

Which of the following identities can be derived from the Pythagorean identity by dividing by sine squared?

1 + 𝑐𝑜𝑡2 (𝜃) = 𝑐𝑠𝑐2 (𝜃) Signup and view all the answers

What can be combined from the Quadrant II and Quadrant IV solutions?

𝜃 = rac{3𝜋}{4} + 𝜋𝑘 Signup and view all the answers

If $𝑐𝑜𝑠(𝜃) ≠ 0$, what is the result of dividing the Pythagorean identity by $𝑐𝑜𝑠2 (𝜃)$?

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

Study Notes

Basic Calculus Module Notes

Course: Basic Calculus
Module: Various Modules (4-8)
Topics: Trigonometry, Unit Circle, Coterminal Angles, Quadrantal Angles, Conversion of degrees to radians, Arc length and area of a sector, Decimal and DMS conversion, Sine, Cosine, Inverse Trigonometric Functions, Polar Coordinate System

Module 4: Trigonometry: Angles in the Unit Circle

Definition of an angle: A set of points determined by two rays having the same endpoint.
Measuring angles: In degrees and radians. 1 radian is the angle formed when the arc length is equal to the radius of the circle. 2π radians = 360°.
Coterminal angles: Angles that have the same initial and terminal sides.
Quadrantal Angles: Angles whose terminal sides lie on the x-axis or y-axis.
Conversion between degrees and radians: Examples provided for converting between degree and radian measures.
Arc length and area of a sector: Formulas given for calculating arc length and area of a sector of a given circle based on the radius and the sector’s angle.
Discusses conversion between decimal and DMS (degrees, minutes, and seconds).

Module 5: The Unit Circle: Sine and Cosine

Unit Circle: A circle centered at the origin with radius 1.
Definition of sine and cosine: The x-coordinate of a point on the unit circle is the cosine of the angle, and the y-coordinate is the sine of the angle.
Finding sine and cosine of specific angles. Examples of finding sine and cosine for angles located in different quadrants.
Reference angle theorem: Used to find the sine or cosine values of angles that are not in the first quadrant.

Module 6: Circular Trigonometric Functions and Identities

Six circular functions: Sine, cosine, tangent, cotangent, secant, cosecant.
Definitions of each function in terms of x and y coordinates on a circle.
Reciprocal and Quotient Identities for each of the functions.
Pythagorean identities: Fundamental relationships between sine, cosine, and other functions.
Example calculations. Shows how to use known identities and formulas to verify or find unknown values of trigonometric functions.

### Module 7: Inverse Trigonometric Functions

Inverse Trig Functions: Used to find the angle when the trigonometric ratio is known.
Inverse of sine, cosine, tangent and how to apply the definitions.

Module 8: Polar Coordinate System

Polar coordinates: Ordered pair (r, θ). r is the distance from the origin (pole) and θ is the angle made with the polar axis.
Converting from rectangular to polar coordinates.
Converting from polar to rectangular coordinates.
Basic polar graphs: Graphs of equations in polar form. Examples are circles, roses, cardioids.

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Description

Explore the essential concepts of trigonometry in this quiz covering angles in the unit circle. Learn about measuring angles in degrees and radians, coterminal angles, quadrantal angles, and how to calculate arc length and area of a sector. This module provides a solid foundation for understanding trigonometric functions and their applications.