Questions and Answers
What is the value of $sin 60° cos 30° + sin 30° cos 60°$?
What is the expression $rac{cos 45}{sec 30 + cosec 30}$ equal to?
What does the expression $rac{5 cos^2 60 + 4 sec^2 30 - tan^2 45}{sin^2 30 + cos^2 30}$ evaluate to?
What is the value of $sec 30°$?
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What is the value of $tan 45°$?
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Study Notes
Trigonometric Values and Calculations
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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 )
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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} )
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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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Description
Solve various trigonometric expressions involving sine, cosine, and secant functions. Evaluate the given expressions and simplify them to find the final answers.
