Trigonometry Problems

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Questions and Answers

What is the value of $sin 60° cos 30° + sin 30° cos 60°$?

  • $0$
  • $1$ (correct)
  • $rac{1}{2}$
  • $rac{ oot{3}}{2}$

What is the expression $rac{cos 45}{sec 30 + cosec 30}$ equal to?

  • $rac{1}{ oot{3}}$
  • $rac{1}{ oot{2}}$ (correct)
  • $rac{1}{2}$
  • $rac{ oot{3}}{2}$

What does the expression $rac{5 cos^2 60 + 4 sec^2 30 - tan^2 45}{sin^2 30 + cos^2 30}$ evaluate to?

  • $5$
  • $2$
  • $4$ (correct)
  • $7$

What is the value of $sec 30°$?

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

What is the value of $tan 45°$?

<p>$1$ (D)</p> Signup and view all the answers

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Study Notes

Trigonometric Values and Calculations

  • Calculate the expression: ( \sin 60° \cos 30° + \sin 30° \cos 60° )

    • Use known values: ( \sin 60° = \frac{\sqrt{3}}{2} ), ( \cos 30° = \frac{\sqrt{3}}{2} ), ( \sin 30° = \frac{1}{2} ), ( \cos 60° = \frac{1}{2} )
    • The expression simplifies to: ( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{4} + \frac{1}{4} = 1 )
  • Evaluate ( \frac{\cos 45°}{\sec 30° + \csc 30°} )

    • Recall the identities: ( \cos 45° = \frac{1}{\sqrt{2}} ), ( \sec 30° = \frac{2}{\sqrt{3}} ), ( \csc 30° = 2 )
    • The expression becomes: ( \frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}} + 2} )
  • Simplify ( \frac{5 \cos^2 60° + 4 \sec^2 30° - \tan^2 45°}{\sin^2 30° + \cos^2 30°} )

    • Utilize the values: ( \cos 60° = \frac{1}{2} ), ( \sec^2 30° = \frac{4}{3} ), ( \tan 45° = 1 )
    • The numerator calculates as: ( 5 \left(\frac{1}{2}\right)^2 + 4 \cdot \frac{4}{3} - 1 = \frac{5}{4} + \frac{16}{3} - 1 )
    • The denominator uses the Pythagorean identity: ( \sin^2 30° + \cos^2 30° = 1 )
    • This results in a simplified ratio for the entire expression.

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