Trigonometric Identities Quiz
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Questions and Answers

What is the Pythagorean Identity?

  • $ an^2( heta) + 1 = rac{1}{ an( heta)}$
  • $ an^2( heta) + rac{1}{ an^2( heta)} = 1$
  • $ an^2( heta) + 1 = rac{1}{ an^2( heta)}$ (correct)
  • $ rac{1}{ an^2( heta)} + 1 = rac{1}{ an^2( heta)}$

    • What is the value of $sin(90^ heta)$?

  • $ rac{ an(90^ heta)}{0}$
  • $0$
  • $1$ (correct)
  • $ an^2(45^ heta)$

    • Which of these is a reciprocal identity?

  • $ an( heta) = rac{ an^2( heta)}{ an( heta)}$
  • $ an( heta) = rac{1}{ an(- heta)}$ (correct)
  • $ an( heta) = rac{ rac{1}{ an(- heta)}}{ an(- heta)}$
  • $ an( heta) = rac{ rac{1}{ an( heta)}}{1}$

    • What does the even-odd identity state about cosine?

    <p>$ an^2(- heta) = an^2( heta)$</p> Signup and view all the answers

    What is the measure of an obtuse angle?

    <p>$90^ heta ext{ to } 180^ heta$</p> Signup and view all the answers

    Which is true for the double angle formula for sine?

    <p>$ an(2 heta) = rac{2 an( heta)}{1 + an^2( heta)}$</p> Signup and view all the answers

    What is the standard position of an angle?

    <p>Measured from the positive x-axis</p> Signup and view all the answers

    How is an acute angle defined?

    <p>$0^ heta &lt; heta &lt; 90^ heta$</p> Signup and view all the answers

    What is the correct half-angle formula for sine?

    <p>$ an rac{ heta}{2} = rac{1}{ an heta}$</p> Signup and view all the answers

    Study Notes

    Trigonometric Identities

    • Basic Identities

      • Pythagorean Identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 )
      • Reciprocal Identities:
        • ( \csc(\theta) = \frac{1}{\sin(\theta)} )
        • ( \sec(\theta) = \frac{1}{\cos(\theta)} )
        • ( \cot(\theta) = \frac{1}{\tan(\theta)} )
      • Quotient Identities:
        • ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
        • ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} )

    • Co-Function Identities

      • ( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) )
      • ( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) )
      • ( \tan(\frac{\pi}{2} - \theta) = \cot(\theta) )

    • Even-Odd Identities

      • Even: ( \cos(-\theta) = \cos(\theta) )
      • Odd:
        • ( \sin(-\theta) = -\sin(\theta) )
        • ( \tan(-\theta) = -\tan(\theta) )

    • Sum and Difference Formulas

      • ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) )
      • ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
      • ( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} )

    • Double Angle and Half Angle Formulas

      • Double Angle:
        • ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
        • ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) )
      • Half Angle:
        • ( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} )
        • ( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} )

    Angles and Their Measures

    • Types of Angles

      • Acute Angle: ( 0^\circ < \theta < 90^\circ )
      • Right Angle: ( \theta = 90^\circ )
      • Obtuse Angle: ( 90^\circ < \theta < 180^\circ )
      • Straight Angle: ( \theta = 180^\circ )
      • Reflex Angle: ( 180^\circ < \theta < 360^\circ )

    • Angle Measurement Units

      • Degrees: Full circle = ( 360^\circ )
      • Radians: Full circle = ( 2\pi ) radians (1 radian ( \approx 57.3^\circ ))
      • Conversion: ( \theta \text{ (degrees)} = \theta \text{ (radians)} \times \frac{180}{\pi} )

    • Standard Position of Angles

      • Angles measured from the positive x-axis.
      • Counterclockwise rotation is positive; clockwise rotation is negative.

    • Reference Angles

      • The acute angle formed by the terminal side of an angle and the x-axis.
      • Useful for determining the values of trigonometric functions in different quadrants.

    • Quadrants

      • I: ( 0^\circ < \theta < 90^\circ ) (All functions positive)
      • II: ( 90^\circ < \theta < 180^\circ ) (Sine positive)
      • III: ( 180^\circ < \theta < 270^\circ ) (Tangent positive)
      • IV: ( 270^\circ < \theta < 360^\circ ) (Cosine positive)

    Trigonometric Identities

    • Basic Identities include the Pythagorean Identity ( \sin^2(\theta) + \cos^2(\theta) = 1 ), foundational for all trigonometric functions.
    • Reciprocal Identities establish relationships between sine, cosine, and their reciprocal functions:
      • ( \csc(\theta) = \frac{1}{\sin(\theta)} )
      • ( \sec(\theta) = \frac{1}{\cos(\theta)} )
      • ( \cot(\theta) = \frac{1}{\tan(\theta)} )
    • Quotient Identities define tangent and cotangent in terms of sine and cosine:
      • ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
      • ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} )

    Co-Function Identities

    • Pivotal relations between sine, cosine, and tangent for angles complementary to ( \theta ):
      • ( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) )
      • ( \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta) )
      • ( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) )

    Even-Odd Identities

    • Even Function: Cosine remains unchanged for negative inputs: ( \cos(-\theta) = \cos(\theta) ).
    • Odd Functions: Sine and tangent change sign for negative inputs:
      • ( \sin(-\theta) = -\sin(\theta) )
      • ( \tan(-\theta) = -\tan(\theta) )

    Sum and Difference Formulas

    • Useful for calculating the sine, cosine, and tangent of angle sums or differences:
      • ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) )
      • ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )
      • ( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} )

    Double Angle and Half Angle Formulas

    • Double Angle Formulas express trigonometric functions of doubled angles:
      • ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
      • ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) )
    • Half Angle Formulas provide expressions for half angles:
      • ( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} )
      • ( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} )

    Angles and Their Measures

    • Types of Angles are classified based on their measures:
      • Acute: ( 0^\circ < \theta < 90^\circ )
      • Right: ( \theta = 90^\circ )
      • Obtuse: ( 90^\circ < \theta < 180^\circ )
      • Straight: ( \theta = 180^\circ )
      • Reflex: ( 180^\circ < \theta < 360^\circ )

    Angle Measurement Units

    • Angles can be measured in degrees and radians:
      • Full circle: ( 360^\circ ) in degrees, ( 2\pi ) radians in radians.
      • Conversion factor: ( \theta \text{ (degrees)} = \theta \text{ (radians)} \times \frac{180}{\pi} ).

    Standard Position of Angles

    • Angles are measured from the positive x-axis.
    • Counterclockwise rotation is positive, while clockwise rotation is negative.
    • Reference Angles: Acute angles with the x-axis, crucial for evaluating trigonometric functions across quadrants.
    • Quadrants:
      • I: ( 0^\circ < \theta < 90^\circ ); all trig functions are positive.
      • II: ( 90^\circ < \theta < 180^\circ ); sine is positive.
      • III: ( 180^\circ < \theta < 270^\circ ); tangent is positive.
      • IV: ( 270^\circ < \theta < 360^\circ ); cosine is positive.

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    Description

    Test your understanding of basic trigonometric identities, including Pythagorean, reciprocal, co-function, and even-odd identities. Delve into sum and difference formulas to ensure mastery of important concepts in trigonometry. Perfect for students preparing for exams or those seeking to strengthen their math skills.

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