Question Video: Using Trigonometric Values of Special Angles to Evaluate Trigonometric Expressions Mathematics • 10th Grade

Evaluate cos² (𝜋/6) − sin² (𝜋/6).

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Video Transcript

Evaluate cos squared 𝜋 over six minus sin squared 𝜋 over six.

We begin by recalling how the angles from zero to two 𝜋 radians can be shown on a quadrant diagram, where 𝜋 radians is equal to 180 degrees. In this question, we need to consider 𝜋 over six radians. Dividing 180 by six, we see that 𝜋 over six radians is equal to 30 degrees. Our expression can therefore be rewritten as cos squared 30 degrees minus sin squared of 30 degrees. We recall that 30 degrees is one of our special angles where the sin of 30 degrees is one-half and the cos of 30 degrees is root three over two.

Substituting these values into our expression, we have root three over two squared minus a half squared. When squaring a fraction, we square the numerator and denominator separately. This means that root three over two squared is equal to three-quarters and one-half squared is one-quarter, noting that root three squared or root three multiplied by root three is three. Subtracting one-quarter from three-quarters gives us two-quarters, and this simplifies to one-half. We can therefore conclude that cos squared 𝜋 over six minus sin squared 𝜋 over six is equal to one-half.

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