Question Video: Using Periodic Identities to Evaluate a Trigonometric Expression Involving Special Angles

Calculate sin 315°cos 45° − cos 120° sin 330°.

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Video Transcript

Calculate sin of 315 degrees multiplied by cos of 45 degrees minus cos of 120 degrees multiplied by sin of 330 degrees.

Before starting this question, we need to remember some of our special angle properties. The sin of 30 degrees is equal to a half. The sin of 45 degrees is equal to one over root two, also written as root two over two. The sin of 60 degrees is equal to root three over two. We can use these along with the sine graph from 0 to 360 degrees to calculate the values of other special angles. In this question, we need to calculate sin of 315 degrees and sin of 330 degrees. The sine graph is wave-shaped with five key points. Sin of zero is equal to zero. Sin of 90 is one. Sin of 180 is zero. Sin of 270 is negative one. And, sin of 360 is equal to zero.

We know that sin of 30 degrees is equal to one-half. 330 degrees is 30 degrees away from 360 degrees. As the graph is symmetrical, we can see that this will be equal to negative one-half. Sin of 330 degrees equals negative one-half. We can use a similar method using 45 degrees and 315 degrees, as 360 minus 45 equals 315. The sin of 315 degrees is equal to negative one over root two.

We can now repeat this process to calculate cos 45 and cos of 120. The three special angles that we need to learn in relation to cos are the cos of 30 degrees equals root three over two, cos of 45 degrees is equal to one over root two, and cos of 60 degrees is equal to one-half. The cos graph is also wave-shaped and symmetrical. However, it starts at one instead of zero. The cos of zero is equal to one. We already know that the cos of 45 is one over root two. Therefore, we need to use the graph to calculate cos of 120.

As cos of 60 degrees is equal to one-half, we can see from the graph that cos of 120 degrees is equal to negative one-half. We now have four values for sin of 315 degrees, cos of 45 degrees, cos of 120 degrees, and sin of 330 degrees. They are negative one over root two, one over root two, negative a half, and negative a half, respectively. Substituting these values into the calculation gives us negative one over root two multiplied by one over root two minus negative a half multiplied by negative a half. When multiplying two fractions, we multiply the numerators together and, separately, multiply the denominators together.

Negative one multiplied by one is equal to negative one, as multiplying a negative number by a positive number gives us a negative answer. Multiplying the square root of 𝑎 by the square root of 𝑎 gives us an answer of 𝑎. This means that the square root of two multiplied by the square root of two is equal to two. We can also think of this as the square root of four, as two multiplied by two is four. We know that the square root of four equals two. Negative one over root two multiplied by one over root two is equal to negative one-half.

We also need to multiply negative one-half by negative one-half. This is equal to one-quarter, as a half multiplied by a half is a quarter and multiplying two negative numbers gives us a positive answer. We are left with negative a half minus one-quarter.

In order to add or subtract fractions, we need to make sure we have a common denominator. Negative a half is the same as negative two-quarters. We’ve multiplied the top and bottom by two. We need to subtract one-quarter from negative two-quarters. As the denominators are now the same, we just need to subtract the numerators. Negative two minus one is equal to negative three. This means that the final answer is negative three-quarters. Sin of 315 degrees multiplied by cos of 45 degrees minus cos of 120 degrees multiplied by sin of 330 degrees is equal to negative three-quarters.

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