Question Video: Evaluating Trigonometric Expressions Involving Special Angles Mathematics

Find the value of 3 sin 30° sin 60° − cos 0° sec 60° + sin 270° cos² 45°.

04:15

Video Transcript

Find the value of three multiplied by the sin of 30 degrees multiplied by the sin of 60 degrees minus the cos of zero degrees multiplied by the sec of 60 degrees plus sin of 270 degrees multiplied by cos squared of 45 degrees.

In order to answer this question, we will need to recall the sine and cosine of our special angles 30, 45, and 60 degrees. We will also need to recall the reciprocal trigonometric identities, specifically the secant of any angle. Finally, we’ll need to find the product of two trigonometric functions as well as squaring the cos of 45 degrees.

The sin, cos, and tan of 30 degrees, 45 degrees, and 60 degrees can be set out in a table as shown. The sine of these three angles are equal to one-half, root two over two, and root three over two, respectively. The cos of 30, 45, and 60 degrees are equal to root three over two, root two over two, and one-half. Finally, the tangent of the three angles equals one over root three, one, and root three. We can substitute the values of sin of 30 degrees, sin of 60 degrees, and the cos of 45 degrees directly into our expression. The first term three multiplied by the sin of 30 degrees multiplied by the sin of 60 degrees is equal to three multiplied by a half multiplied by root three over two. This is equal to three root three over four.

From the graphs of 𝑦 equals sin 𝜃 and 𝑦 equals cos 𝜃, we see that the cos of zero degrees is equal to one and the sin of 270 degrees is equal to negative one. The sec of any angle 𝜃 is the reciprocal of the cos of the angle such that the sec of 𝜃 is equal to one over the cos of 𝜃. This means that the sec of 60 degrees is equal to one over the cos of 60 degrees. As the cos of 60 degrees is equal to one-half, the sec of 60 degrees is equal to two.

The second term in our expression, the cos of zero degrees multiplied by the sec of 60 degrees, is therefore equal to one multiplied by two. This is equal to two. Finally, the third term, the sin of 270 degrees multiplied by cos squared 45 degrees, is equal to negative one multiplied by root two over two squared. Root two over two squared is equal to two over four, as we simply square the numerator and denominator. This simplifies to one-half, which we need to multiply by negative one. The third term is, therefore, equal to negative one-half.

Substituting our three values into the original expression, we have three root three over four minus two plus negative one-half. Negative two plus negative one-half is the same as negative two and a half, which is equal to negative five over two. We can then multiply the numerator and denominator of our second fraction by two. This gives us three root three over four minus 10 over four. As the denominators are the same, we can subtract the numerators, giving us negative 10 plus three root three over four. This is the value of the original expression.

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