Trigonometry Basics and Unit Circle
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Questions and Answers

What represents the radian measure of an angle in relation to the unit circle?

  • The degree measure of the angle subtended by the arc
  • The area of the sector formed by the angle
  • The circumference of the circle
  • The length of the arc corresponding to the angle (correct)

How are positive and negative real numbers represented in relation to the unit circle?

  • Positive real numbers are represented anticlockwise, and negative real numbers clockwise (correct)
  • Both positive and negative real numbers are represented in the anticlockwise direction
  • Positive real numbers are represented clockwise, and negative real numbers anticlockwise
  • Positive real numbers are represented in the vertical direction, and negative ones in the horizontal

What is the relationship between radians and degrees for a full circle?

  • 90° = π/2 radians
  • 180° = π radians
  • Both A and B are correct (correct)
  • 360° = 2π radians

What is the approximate value of 1 radian in degrees?

<p>57° 16′ (C)</p> Signup and view all the answers

What is the radian measure equivalent of 1° expressed in radians?

<p>0.01746 radian (A)</p> Signup and view all the answers

What is the primary purpose of trigonometry as described?

<p>To solve geometric problems involving triangles (A)</p> Signup and view all the answers

Which of the following applications does NOT typically use trigonometry?

<p>Calculating chess moves (B)</p> Signup and view all the answers

What defines a positive angle?

<p>Angle formed by an anticlockwise rotation (D)</p> Signup and view all the answers

What is the vertex of an angle?

<p>The rotation point of the ray (C)</p> Signup and view all the answers

Which of the following measures is NOT commonly used for measuring angles?

<p>Circumferential measure (B)</p> Signup and view all the answers

How is an angle defined mathematically?

<p>As the measure of rotation of a ray (D)</p> Signup and view all the answers

What is trigonometry primarily concerned with?

<p>Measuring the sides and angles of triangles (A)</p> Signup and view all the answers

Which is a result of the generalization of trigonometric ratios?

<p>Trigonometric functions (A)</p> Signup and view all the answers

What is the relationship between the values of sin x and sin (–x)?

<p>sin (–x) = –sin x (C)</p> Signup and view all the answers

What is the range of values for cos x based on the unit circle?

<p>–1 to 1 (A)</p> Signup and view all the answers

What is the value of tan(90°)?

<p>undefined (A)</p> Signup and view all the answers

Which of the following statements is true regarding the trigonometric function sec x?

<p>sec x is equal to 1/cos x. (B)</p> Signup and view all the answers

In which quadrant do both sine and cosine values of x remain positive?

<p>First quadrant (B)</p> Signup and view all the answers

What is the measure of one degree in terms of minutes?

<p>60' (C)</p> Signup and view all the answers

Which of the following angles is equivalent to –30°?

<p>330° (D)</p> Signup and view all the answers

What is the angle in radians of one full revolution?

<p>2π (A)</p> Signup and view all the answers

If an arc of length 3 units subtends an angle of θ radians in a circle of radius 3 units, what is θ?

<p>1 radian (B)</p> Signup and view all the answers

What is the radian measure equivalent to 90 degrees?

<p>$\frac{\pi}{2}$ (D)</p> Signup and view all the answers

How many seconds are there in one degree?

<p>3600″ (C)</p> Signup and view all the answers

Which formula correctly converts degrees to radians?

<p>Radian measure = Degree measure $\times \frac{\pi}{180}$ (A)</p> Signup and view all the answers

Convert 180 degrees into radians.

<p>$\pi$ (B)</p> Signup and view all the answers

What is the relationship between radians and arc length?

<p>Arc length equals the radius multiplied by the angle in radians (A)</p> Signup and view all the answers

If $\frac{3\pi}{4}$ radians is the measure of an angle, what is its equivalent in degrees?

<p>135° (B)</p> Signup and view all the answers

What is the measure of an angle in radians that subtends an arc of length 5 units in a circle with radius 5 units?

<p>1 radian (C)</p> Signup and view all the answers

Which statement about radians and degrees is true?

<p>1 radian is approximately 57.3° (B)</p> Signup and view all the answers

How many radians are in 270 degrees?

<p>$\frac{3\pi}{2}$ (B)</p> Signup and view all the answers

What is the degree measure of an angle that is $\frac{5\pi}{6}$ radians?

<p>150° (A)</p> Signup and view all the answers

Which of the following angles represents a full rotation?

<p>360° (B)</p> Signup and view all the answers

What is the conversion of 40° 20′ into radians?

<p>$\frac{121\pi}{540}$ (A)</p> Signup and view all the answers

What values of x make sin x equal to 0?

<p>x = nπ where n is any integer (B)</p> Signup and view all the answers

When does cos x equal 0?

<p>x = ±(2n + 1)π/2 where n is any integer (D)</p> Signup and view all the answers

What is the expression for tan x in terms of sine and cosine?

<p>tan x = sin x / cos x (D)</p> Signup and view all the answers

Which of the following identities is correct?

<p>1 + tan^2 x = sec^2 x (A)</p> Signup and view all the answers

For which value of x is cos x equal to -1?

<p>x = 3π (C)</p> Signup and view all the answers

Which of the following statements about cosec x is true?

<p>cosec x = 1/sin x, where x ≠ nπ (B)</p> Signup and view all the answers

Which statement about periodicity of sine and cosine functions is true?

<p>sin(x + 2π) = sin x and cos(x + 2π) = cos x (C)</p> Signup and view all the answers

Flashcards

Initial Side of an Angle

The original position of a ray before rotation.

Terminal Side of an Angle

The final position of a ray after rotation.

Vertex of an Angle

The point where the initial and terminal sides meet.

Angle

A measure of rotation of a ray around its initial point.

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Positive Angle

An angle created by rotating a ray counter-clockwise.

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Negative Angle

An angle created by rotating a ray clockwise.

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Full Circle (360 Degrees)

One complete revolution of a ray back to its original position.

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Degree Measure

A common unit for measuring angles, where a full circle is divided into 360 degrees.

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Full Circle

One complete revolution of a ray around its initial point.

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Radian Measure

A unit for measuring angles based on the length of an arc in a circle.

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Unit Circle

A unit circle is a circle with a radius of 1 unit. It's used to visually represent angles and their relationships to real numbers.

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Real Numbers and Radian Measures

Real numbers can be represented on a number line, and when wrapped around a unit circle, they correspond to specific radian measures.

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Radian and Degree Relationship

A radian measure is related to a degree measure by the conversion: 2π radians = 360°. This means one full rotation is represented by both 2π radians and 360 degrees.

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Converting Radians and Degrees

The relationship between degree and radian measures allows us to convert between the two units. There are approximately 57.3 degrees in one radian, and approximately 0.01746 radians in one degree.

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Sine function sign?

The sine function is positive in the first and second quadrants, negative in the third and fourth quadrants.

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Cosine function sign?

The cosine function is positive in the first and fourth quadrants, negative in the second and third quadrants.

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Tangent function sign?

The tangent function is positive in the first and third quadrants, negative in the second and fourth quadrants.

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Reciprocal trigonometric functions

The cosecant, secant, and cotangent functions are the reciprocals of the sine, cosine, and tangent functions, respectively.

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Range of sine and cosine

For any angle x, -1 ≤ sin(x) ≤ 1 and -1 ≤ cos(x) ≤ 1. This means the sine and cosine values always fall within the range of -1 to 1.

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Angle Measures

The relationship between degree measure and radian measure for angles commonly used in trigonometry. Radians represent the arc length of a circle with a radius of one unit. For example, a 360 degree angle corresponds to a full circle arc length of 2π radians.

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Angle Notation

A convention used in mathematics to denote angles. For example, θ° represents an angle with a degree measure of θ, while β represents an angle with a radian measure of β.

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Conversion Factor (π=180°)

π radians correspond to 180 degrees. This conversion acts as a bridge between degrees and radians, allowing us to convert between the two units.

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Converting Degrees to Radians

Calculating the radian measure of an angle by multiplying its degree measure by π/180.

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Converting Radians to Degrees

Calculating the degree measure of an angle by multiplying its radian measure by 180/π.

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Angle Conversion

This is the process of converting an angle from one unit (degrees or radians) to another. It allows us to perform calculations and understand angles expressed in different units.

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Radian Omission Convention

Any angle expressed in radians, the 'radian' unit is often omitted. For example, π/4 is understood to be π/4 radians.

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Degrees, Minutes, Seconds

A common way of expressing angles using degrees, minutes (′), and seconds (″). Each degree is divided into 60 minutes, and each minute is divided into 60 seconds.

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Periodicity of Sine and Cosine

The value of both sine and cosine functions remains unchanged when the angle is increased or decreased by any multiple of 2π. This is because a complete revolution around a circle brings you back to the same starting point.

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When is sin(x) = 0?

For any angle x, the sine of that angle is equal to 0 when x is an integral multiple of π (e.g., 0, π, 2π, -π, etc.).

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When is cos(x) = 0?

For any angle x, the cosine of that angle is equal to 0 when x is an odd multiple of π/2 (e.g., π/2, 3π/2, 5π/2, etc.).

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Cosecant Function (csc x)

The cosecant function (csc x) is the reciprocal of the sine function (sin x). So, csc x = 1/sin x. It's undefined when sin x = 0, which occurs at integral multiples of π.

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Secant Function (sec x)

The secant function (sec x) is the reciprocal of the cosine function (cos x). So, sec x = 1/cos x. It's undefined when cos x = 0, which occurs at odd multiples of π/2 .

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Tangent Function (tan x)

The tangent function (tan x) is defined as the ratio of the sine function (sin x) to the cosine function (cos x). So, tan x = sin x/cos x. It's undefined when cos x = 0, which occurs at odd multiples of π/2.

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Cotangent Function (cot x)

The cotangent function (cot x) is the reciprocal of the tangent function (tan x). So, cot x = 1/tan x = cos x/sin x. It's undefined when sin x = 0, which occurs at integral multiples of π.

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Fundamental Trigonometric Identity

The fundamental trigonometric identity states that for any angle x: sin² x + cos² x = 1. It's a basic relationship that holds true for all angles.

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Study Notes

Trigonometric Functions

  • Introduction:
    • Trigonometry is derived from Greek words meaning "measuring the sides of a triangle".
    • It's used to solve geometric problems involving triangles in various fields such as navigation, surveying, and seismology.
    • It helps in designing electric circuits, describing atomic states, predicting tides, and analyzing musical tones.
  • Angles:
    • Angle is the measure of rotation of a ray about its starting point (vertex).
    • The starting position of the ray is the initial side.
    • The final position of the ray after rotation is the terminal side.
    • Positive angles are formed by counterclockwise rotation.
    • Negative angles are formed by clockwise rotation.
  • Degree Measure:
    • A degree (°) is 1/360th of a complete revolution.
    • A degree is further divided into 60 minutes (') and a minute into 60 seconds (").
    • 1° = 60', 1' = 60"
  • Radian Measure:
    • An angle of 1 radian is subtended at the centre of a unit circle by an arc of length 1 unit.
    • A complete revolution corresponds to 2π radians.
    • In a circle of radius r, an arc of length l will subtend an angle of θ = l/r radians.
  • Relationship between Degrees and Radians:
    • π radians = 180°
    • 1 radian ≈ 57.3°
    • 1° ≈ 0.01746 radians
  • Trigonometric Functions:
    • Trigonometric functions are defined for all real numbers (angles).
    • The domain of sine and cosine functions is the set of all real numbers, with range [-1, 1].
    • The reciprocal functions of sine and cosine are cosecant and secant
    • The reciprocal functions of tangent and cotangent are cotangent and tangent respectively.

Trigonometric Functions of Special Angles

  • Tables of trigonometric values for special angles (0°, 30°, 45°, 60°, 90°) are provided (angles in degrees and radians).
  • These values are the same as those derived from right-angled triangles.

Relation Between Degree and Radian Measure

  • Given an angle in degrees, one can convert in to radians or vise versa using the relationship π radians = 180°.

Trigonometric Functions of Sum and Difference of Two Angles

  • Formulas for trigonometric functions of the sum and difference of two angles (x and y) are provided.
  • Identities involving cos (x+y), cos (x-y), sin (x+y), sin (x-y) are included.
  • Examples demonstrate calculating trigonometric values given angles.

Trigonometric Functions: Miscellaneous Examples

  • Example problems demonstrate the application of identities to find the values of trigonometric functions in special cases and quadrants.

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Description

This quiz covers fundamental concepts of trigonometry and its relationship with the unit circle. You'll explore angle measures, the significance of radians and degrees, and the applications of trigonometric functions. Test your understanding of these key topics in trigonometry!

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