Trigonometric Identities Quiz
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Questions and Answers

What defines an identity in trigonometric equations?

  • An equation that remains true for all values of the variable. (correct)
  • A conditional equation that holds true for some values.
  • An equation that is only true for certain intervals.
  • An equation that has at least one solution.

Which technique is NOT suggested for verifying trigonometric identities?

  • Assume both sides are equal and simplify. (correct)
  • Factor out the greatest common factor.
  • Rewrite the complicated side in terms of sines and cosines.
  • Combine fractional expressions using the least common denominator.

If an equation does not satisfy some values in its domain, what is it classified as?

  • An identity.
  • A conditional equation. (correct)
  • An absolute equation.
  • A universal equation.

Which of the following is NOT a fundamental identity type mentioned?

<p>Exponential Identities. (B)</p> Signup and view all the answers

When verifying trigonometric identities, which strategy is recommended?

<p>Transform each side step-by-step independently. (A)</p> Signup and view all the answers

What is the radian measure of a 90 degree angle?

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

What is the sine of 180 degrees?

<p>0 (B)</p> Signup and view all the answers

What are the coordinates of the point on the unit circle that corresponds to an angle of 3π/4 radians?

<p>(-√2/2, √2/2)</p> Signup and view all the answers

What is the tangent of 270 degrees?

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

Which of the following angles has a cosine value of -1/2?

<p>240 degrees (C)</p> Signup and view all the answers

What is the radian measure of 30 degrees?

<p>π/6 (D)</p> Signup and view all the answers

Flashcards

Trigonometric Identity

An equation that is true for all values of the variable within its domain.

Verifying a Trigonometric Identity

Showing that two trigonometric expressions are equal for all valid inputs.

Fundamental Trig Identities

Basic trigonometric relationships between sine, cosine, tangent, etc.

Strategy for verifying Identities

Techniques used to transform one side of the equation to match the other.

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Conditional Equation

An equation that is true only for specific values (not all) of the variable.

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Unit Circle

A circle with a radius of 1 centered at the origin of a coordinate plane, where points represent angles and their trigonometric values.

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Trigonometric Functions

Functions like sine, cosine, and tangent that relate angles to the ratios of sides in a right triangle.

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Sine (sin)

The y-coordinate of a point on the unit circle representing an angle.

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Cosine (cos)

The x-coordinate of a point on the unit circle representing an angle.

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Tangent (tan)

The ratio of sine to cosine (sin/cos), representing the slope of the line from the origin to a point on the unit circle.

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Key Points (Unit Circle)

Points on the unit circle corresponding to angles like 0°, 30°, 45°, 60°, 90°, etc., representing important trigonometric values.

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Radians

A unit for measuring angles, defined by the ratio of the arc length to the radius of a circle.

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2π radians

The equivalent of a full circle in radians.

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Degrees to Radians

Converting angles from degrees to radians, using the conversion factor π/180.

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Radians to Degrees

Converting angles from radians to degrees, using the conversion factor 180/π.

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Pythagorean Identity

The relationship sin²θ + cos²θ = 1, true for all angles θ.

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Reciprocal Trig Functions

Functions like cosecant (csc), secant (sec), and cotangent (cot) which are reciprocals of sine, cosine, and tangent respectively.

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Reference Angle

The acute angle formed between the terminal side of an angle and the x-axis.

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Quadrant Signs (sin, cos, tan)

The signs of sine, cosine, and tangent vary in different quadrants based on the reference angle.

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Periodic Functions

Functions that repeat their values over regular intervals.

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Trigonometric Period

The length of the interval over which a trigonometric function repeats.

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Angle and Trig Ratios

The relationship between an angle and its corresponding sine, cosine, and tangent values.

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Solving Trig Equations

Finding the values of the unknown angles in a trigonometric equation by manipulating the equation.

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Graphing Trig Functions

Representing trigonometric functions visually on a graph.

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Verifying Identities

Proving that two trigonometric expressions are equivalent for all valid inputs.

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Unit Circle Coordinates (30° and 45°)

The coordinates of points on the unit circle corresponding to multiples of 30° and 45° angles.

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Symmetry Properties (Trig Functions)

Relationships between trigonometric functions that allow for quicker calculations by relating values of different angles.

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Inverse Trig Functions

Functions that find the angle corresponding to a given trigonometric value.

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Arc Length

The distance along the curved part of a circle.

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Radian Measure (Arc Length/Radius)

The ratio of the arc length to the radius of a circle, which gives the angle in radians.

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Trigonometric Applications (Real-world)

Practical applications of trigonometric functions in fields like physics, engineering, and navigation.

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Study Notes

Trigonometric Identities

  • Trigonometric identities are equations that are true for all values of the variable in the domain of the equation
  • An equation that is not an identity is called a conditional equation. This occurs when some values of the variable in the domain of the equation do not satisfy the equation

Fundamental Trigonometric Identities

  • Reciprocal Identities

    • sin x = 1/csc x
    • cos x = 1/sec x
    • tan x = 1/cot x
    • csc x = 1/sin x
    • sec x = 1/cos x
    • cot x = 1/tan x

  • Quotient Identities

    • tan x = sin x/cos x
    • cot x = cos x/sin x

  • Pythagorean Identities

    • sin² x + cos² x = 1
    • 1 + tan² x = sec² x
    • 1 + cot² x = csc² x

Even-Odd Identities

  • sin(-x) = -sin x
  • cos(-x) = cos x
  • tan(-x) = -tan x
  • csc(-x) = -csc x
  • sec(-x) = sec x
  • cot(-x) = -cot x

Guidelines for Verifying Trigonometric Identities

  • Work with each side of the equation independently. Start with the more complicated side and work towards the other side in a series of steps.
  • Analyze the identity and look for opportunities to apply fundamental identities.
  • Techniques to utilize include:
    • Rewrite the more complicated side in terms of sines and cosines
    • Factor out the greatest common factor
    • Separate a single-term quotient into two terms
    • Combine fractional expressions using the least common denominator
    • Multiply the numerator and the denominator by a binomial factor that appears on the other side of the identity

Additional Trigonometric Identities

  • Sum and Difference Identities

    • sin(a + b) = sin a cos b + cos a sin b
    • sin(a - b) = sin a cos b - cos a sin b
    • cos(a + b) = cos a cos b - sin a sin b
    • cos(a - b) = cos a cos b + sin a sin b
    • tan(a + b) = (tan a + tan b) / (1 - tan a tan b)
    • tan(a - b) = (tan a - tan b) / (1 + tan a tan b)

  • Double Angle Identities

    • sin 2a = 2 sin a cos a
    • cos 2a = cos² a - sin² a = 2 cos² a - 1 = 1 - 2 sin² a
    • tan 2a = 2 tan a / (1 - tan² a)

  • Half Angle Identities

    • sin (a/2) = ±√((1-cos a)/2)
    • cos (a/2) = ±√((1+cos a)/2)
    • tan (a/2) = ±√((1 - cos a) / (1 + cos a)) = sin a / (1 + cos a) = (1 - cos a) / sin a

Example Identities to Verify

  • csc θ cot θ = sin θ / (1 + cos θ)
  • sec x cot x = csc x
  • cos² x - cos x sin² x = cos³ x
  • (1 + sin x) / cos x = sec x + tan x

Seatwork Examples (Identities to Prove)

  • 2 sin² θ = 2 - 2 cos² θ
  • csc θ sec θ - cot θ = tan θ

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Trigonometric Identities PDF

Description

Test your knowledge of trigonometric identities with this quiz. Explore various reciprocal, quotient, and Pythagorean identities as well as even-odd identities. Understand the key concepts that are crucial for solving trigonometric equations.

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