Trigonometric Identities and Equations Quiz
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Questions and Answers

The ______ identity states that csc x = 1 / sin x.

reciprocal

The ______ identity states that tan x = sin x / cos x.

quotient

To solve for sin x in this case, we must use the ______ ______ identity.

Pythagorean Theorem

The ______ identity states that cos (-x) = cos x.

<p>even-odd</p> Signup and view all the answers

The ______ identity states that cos (-π/2 + θ) = sin θ.

<p>cofunction</p> Signup and view all the answers

When simplifying the expression, we can substitute sec x with 1 / ______.

<p>cos x</p> Signup and view all the answers

We can then simplify the expression using the ______ identity.

<p>Pythagorean</p> Signup and view all the answers

To prove a trigonometric identity, we must begin with one side of the equation and manipulate it to match the ______ side.

<p>other</p> Signup and view all the answers

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What is the value of csc x if sin x = 2√5 / 5?

<p>√5 / 2 (D)</p> Signup and view all the answers

What is the value of cos a if tan a = √2 / 2 and sin a = -√3 / 2?

<p>-√6 / 2 (C)</p> Signup and view all the answers

What is the value of sin x if cot x = -√3 / 2 and cos x < 0?

<p>2√7 / 7 (B)</p> Signup and view all the answers

What is the value of cot θ if cos θ = -7 / 12 and sin θ < 0?

<p>-7√95 / 95 (D)</p> Signup and view all the answers

What is the value of tan (-x) if cot (-x) = -1.84?

<p>-0.54 (A)</p> Signup and view all the answers

What is the value of cos (-π/2 + θ) if sin θ = -0.57?

<p>-0.57 (B)</p> Signup and view all the answers

Which of the following is a simplified form of sec x * sin² x + cos x?

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

Which of the following is a simplified form of csc ß - sin ß / sin ß?

<p>cos² ß / sin ß (D)</p> Signup and view all the answers

Flashcards

Cosecant Definition

Csc x is the reciprocal of sin x.

Finding cosine from tangent

Cos a can be found using sin a and tan a.

Pythagorean Identity

sin² x + cos² x = 1 applies to trigonometric functions.

Cotangent Definition

Cot θ is the ratio of cos θ to sin θ.

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Even-Odd Identities

Tan(-x) = -tan(x) due to odd behavior.

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Cofunction Identity

cos(-π/2 + θ) = sin θ.

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Secant Simplification

Sec x = sin² x/cos x + cos x simplifies to 1/cos x.

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Cosecant Derivation

csc ß - sin ß/sin ß simplifies to cos² ß/sin ß.

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Csc x from sin x

If sin x = 2√5 / 5, then csc x = √5 / 2.

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Finding cos a from tan and sin

Cos a can be calculated as cos a = (-√3 / 2) / (√2 / 2).

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Cotangent and sine relationship

Cot x = cos x / sin x relates cotangent to sine and cosine.

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Find sin x using cot x

Using cot x = -√3 / 2, find sin x = 2√7 / 7.

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Cotangent value from cosine

Cot θ = -7 / √95 when cos θ = -7 / 12 and sin θ < 0.

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Tan of negative angle

Tan (-x) can be calculated as tan (-x) = -1 / 1.84.

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Cofunction identity with sine

cos(-π/2 + θ) = sin θ relates cos of a combination angle to sine.

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Simplifying sin with cotangent

sin y + sin y * cot² y / csc y simplifies to csc y.

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Study Notes

Trigonometric Identities and Equations

  • Reciprocal and Quotient Identities: Used for questions 1 & 2

    • csc x = 1/sin x
    • sec x = 1/cos x
    • tan x = sin x/cos x
    • cot x = cos x/sin x
  • Pythagorean Theorem Identities: Used for questions 3 & 4

    • sin² x + cos² x = 1
    • 1 + tan² x = sec² x
    • 1 + cot² x = csc² x
  • Cofunction and Even-Odd Identities: Used for questions 5 & 6

    • These identities relate trigonometric functions of complementary angles and functions of negative angles.
    • Examples are needed to detail the cofunction and even-odd identities
    • Specific examples (e.g., sin(90° - θ) = cos θ, cos(-x) = cos x, etc.) are needed to provide specific formulas.
  • Simplifying Trigonometric Expressions (Questions 7-10):

    • Involves manipulating trigonometric expressions using identities to simplify. Example: Simplifying sec x sin² x + cos x / cos x
    • Example problem 8, simplifying csc β - sin β / sin β.
    • An example of a simplified expression: cos²β / sinβ.
  • Proving Trigonometric Identities (Questions 11-12): - Involve showing that one trigonometric expression is equal to another using established identities. - Example problem 12, proving that sin a / (1 - cos a) = csc a + cot a - Additional examples of identities are needed to clarify this section.

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Description

Test your knowledge on trigonometric identities and equations through a variety of questions. This quiz covers reciprocal, quotient, Pythagorean, cofunction, and even-odd identities, as well as simplifying expressions and proving identities. Master these essential concepts to excel in trigonometry.

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