Before going to our main topic (sine 60 degrees), let’s recall some important points for a better understanding of the concept. We have already studied triangles, and in particular, right triangles. Especially in the concept of trigonometry, the primary ratios like sine, cosine, tangent are normally used to find the angles, distances or heights, and lengths of the right-angled triangle.
Now, consider a right triangle ABC as shown in the figure.
Here, ∠ CAB is an acute angle. Note the position of side BC w.r.t.∠ A. It is the opposite side to ∠ A. AC is the hypotenuse of the right triangle, and the side AB is part of ∠ A. We further work certain ratios involving the sides of a right triangle. These ratios are known as trigonometric ratios.
The trigonometric ratios of ∠ A in right triangle ABC are defined as follows:
Similarly, cosec A, sec A, cot A can also be determined.
Now we are clear with the introduction part, let us now calculate the trigonometric ratio of sine 60o.
Consider an equilateral triangle ABC. Since, each angle in an equilateral triangle is 60 o, ∠ A = ∠ B = ∠ C = 60o.
Draw a perpendicular AD from A to side BC, as shown in the figure.
Now, Δ ABD is a right triangle, right-angled at D with
∠ ABD = 60o.
As we know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle. So, let us suppose that AB = 2a.
Let us know all the trigonometric ratios of sine 0 o, 30 o, 45 o, 60 o and 90o.
Sin 0o = 0
Sin 30o = 1/2
Sin 45o = 1/√2
Sin 60o = √3/2
Sin 90o= 1
We can simply memorize the trigonometric ratios of cosine by writing the values of sine from bottom to top, i.e.,
|Cos 0o||=||Sin 90o||=||1|
|Cos 30o||=||Sin 60o||=||√3/2|
|Cos 45o||=||Sin 45o||=||1/√2|
|Cos 60o||=||Sin 30o||=||1/2|
|Cos 90o||=||Sin 0o||=||0|
PROBLEM – 1: Sin 60o Cos30o + Sin30o Cos60o
SOLUTION: We know that,