Find the value of sin of 180 degrees minus 𝑥 plus tan of 360 degrees minus 𝑥 plus seven multiplied by sin 270 degrees minus 𝑥 given sin 𝑥 is equal to three-fifths where 𝑥 is greater than zero but less than 90 degrees.
We can use the fact that sin 𝑥 is equal to three-fifths to calculate the value of cos 𝑥 and tan 𝑥. As 𝑥 is between zero and 90 degrees, we know that sin 𝑥, cos 𝑥, and tan 𝑥 will all be positive. Creating and then using our knowledge of right-angled trigonometry, we know that the opposite is equal to three and the hypotenuse is equal to five. This is one of our special Pythagorean triangles. So the adjacent is four. If sin 𝑥 is equal to three-fifths, then cos 𝑥 will be equal to four-fifths, the adjacent over the hypotenuse. Tan 𝑥 in this first quadrant is equal to three-quarters, as it is the opposite over the adjacent.
Our next step is to use our knowledge of compound angles to rewrite our expression. Sin of 𝐴 minus 𝐵 is equal to sin 𝐴 cos 𝐵 minus cos 𝐴 sin 𝐵. We can, therefore, rewrite sin of 180 minus 𝑥 as sin of 180 multiplied by cos 𝑥 minus cos of 180 multiplied by sin 𝑥. We know that the sin of 180 degrees is equal to zero. The cos of 180 degrees is equal to negative one. Zero multiplied by cos 𝑥 is equal to zero. And minus negative one sin 𝑥 is equal to positive sin 𝑥. We were told in the question that sin 𝑥 is equal to three-fifths. Therefore, the first term simplifies to three-fifths.
We can use the same formula to simplify the third term, seven multiplied by sin of 270 degrees minus 𝑥. This is equal to seven multiplied by sin 270 degrees multiplied by cos 𝑥 minus cos of 270 degrees multiplied by sin 𝑥. Sin of 270 degrees is equal to negative one. Cos of 270 degrees is equal to zero. Therefore, the second term in the bracket or parentheses equals zero. This term simplifies to negative seven cos 𝑥. We know that cos 𝑥 is equal to four-fifths. This means we need to multiply negative seven by four-fifths. This is equal to negative 28 over five or negative twenty-eight fifths. The third term in our expression is equal to negative twenty-eight fifths.
Let’s now consider the second term. Tan of 𝐴 minus 𝐵 is equal to tan 𝐴 minus tan 𝐵 over one plus tan 𝐴 multiplied by tan 𝐵. We can use this formula to simplify the second term. Tan of 360 minus 𝑥 is equal to tan 360 minus tan 𝑥 divided by one plus tan 360 multiplied by tan 𝑥. The tan of 360 degrees is equal to zero. The expression simplifies to negative tan 𝑥 over one, which is just negative tan 𝑥. As tan 𝑥 is equal to three-quarters negative tan 𝑥 is equal to negative three-quarters. The second term in our expression is negative three-quarters.
We now have values for all three terms. The value of sin of 180 minus 𝑥 plus tan of 360 minus 𝑥 plus seven multiplied by sin of 270 minus 𝑥 is equal to three-fifths minus three-quarters minus twenty-eight fifths. Three-fifths minus twenty-eight fifths is equal to negative twenty-five fifths. This is equal to negative five. Our value simplifies to negative five minus three-quarters. This is equal to negative five and three-quarters or negative twenty-three quarters. Either one of these answers or the decimal negative 5.75 is correct.