State the exact value of sine of 60 degrees.
There are a table of trigonometric values that we must know for our exams. And there is a couple of different ways to remember it. Let’s look at one of them. The values we need to know are sine, cosine, and tan of zero, 30, 45, 60, and 90 degrees. Sin of zero is zero and sin of 90 degrees is one. It’s then the opposite for cosine: cos of zero is one and cos of 90 degrees is zero.
We then write one, two, and three under the values for sin 30, sin 45, and sin 60 and reverse that for cosine to get three, two, one. We make the denominator of these six values two and we find the square root for the numerator for each one. Remember the square root of one is simply one. So we don’t actually need to write the square root symbol here. Unfortunately, for tan, there are no such special tricks. We just need to learn them. Notice that there is no value for tan of 90 degrees.
Once we know this table, we can then use it to find exact values of the trigonometric functions. Sin of 60 degrees in our table is the square root of three over two.
Using the diagram below, work out the value of 𝑥.
Now, let’s consider the triangle. It’s a right-angled triangle with a given angle of 60 degrees. That means we need to use right angle trigonometry to help us find the length of the missing side. We’ll label the triangle first.
The hypotenuse is the longest side of the triangle. It’s found by looking for the side that’s opposite the right angle. The opposite side is the side opposite the given angle. And the adjacent is the remaining side. It will always be the side immediately next to the given angle.
In this triangle, we know the length of the adjacent and we want to find the length of the hypotenuse. That means we use cosine. Cos of 𝜃, where 𝜃 is the angle, is equal to the adjacent over the hypotenuse. Substituting the values from our triangle gives us cos of 60 degrees is equal to four over 𝑥.
To solve this equation, we’ll start by multiplying both sides of the equation by 𝑥 to give us 𝑥 multiplied by cos of 60 degrees is equal to four. Next, we’ll divide both sides of the equation by cos of 60 degrees. That gives us 𝑥 is equal to four divided by cos of 60 degrees, which would normally typed into our calculator.
However, let’s refer back to our table and we can see that cos of 60 degrees is equal to a half. So we can replace cos of 60 with a half in our equation. Remember to divide by a fraction, we find the reciprocal of the second fraction and change the divides to a times. That gives us 𝑥 is equal to four multiplied by two over one.
Another way of saying that is four over one multiplied by two over one. Multiplying the numerators gives us eight and multiplying the denominators gives us one. Since eight over one is just eight, we found the value of 𝑥 to be eight.