Trigonometry-Identities & Equations-Trigonometric Ratios For Compound Angles

Trigonometry is a branch of mathematics that is mainly used to deal with angles, or more specifically, this branch works with the side and angle of right triangles. In trigonometry, there are six trigonometric ratios sine, cosine, tangent, cotangent, cosecant, and secant. A triangle can be a right angle triangle or a right triangle only if there is an angle of 90 degrees. The sides of the triangle are given as, longest side – Hypotenuse, the other two sides are named as adjacent and opposite sides according to the angles placed.

These ratios can be given for an angle as follows:

Compound Angles

When we talk about the compound angle, we can define them as the sum of two or more angles. Using the trigonometric identities, we can denote compound angles using trigonometry functions. This trigonometry section mainly works on functions like (A+B) and (A-B), where A and B are the values of angles.

In the above, we have seen that since, cosine, tangent, cotangent, cosecant, and secant are the trigonometry ratio, and compound angles are algebraic combinations of two or more angles. When we use trigonometry ratios with compound angles, trigonometric ratios for compound angles come into the picture.

This topic involves many formulas for the trigonometric ratio of compound angles. Some of the formulas are as follows:

1. sin(c+d)=sin(c)cos(d)+cos(c)sin(d)
2. sin(c-d)=sin(c)cos(d)-cos(c)sin(d)
3. cos(c+d)=cos(c)cos(d)-sin(c)sin(d)
4. cos(c-d)=cos(c)cos(d)+sin(c)sin(d)
5. tan(c+d)=tan(c)+tan(d)1-tan(c)tan(d)
6. tan(c-d)=tan(c)-tan(d)1+tan(c)tan(d)

The value of the trigonometry ratios can be found in the following tables. 

Here in the above list, we can find values of some of the standard trigonometric ratios. In most of the problems related to compound angles and trigonometry, we can find trigonometric ratios. 

In the above, we have discussed some of the formulas of trigonometry ratio of compound angle. The formulas can be used in various complex examples and equations to make them easy. Let’s explain them more using some set of problems.

Examples 

In this section, we will look at some of the problems of trigonometry that can be resolved using the trigonometric ratio of compound angles.

1. Problem: What is the value of cos 15°?

Solution:

In the above, the given value of the angle can be written as follows. 

15° = 60° – 45°

That means 

cos 15° = cos(60°-45°)

Now from the above-discussed formula, we know that. 

cos(c-d)=cos(c)cos(d)+sin(c)sin(d)

We can compare the above rewriting of cos 15° as following 

• 60° = c
• 45° = d

Let’s put these values in the formula.

Cos 15° =  cos 60°.cos 45° + sin 60°.sin 45°

Let’s take the values from the above-given table for values of trigonometric ratios.

Cos 15° = (1/√2) ⋅ (√3/2) + (1/√2) ⋅ (1/2)

= (√6/4)  +  (√2/4)

= (√6 + √2)/4

Here from the above outcome, we can say that. 

Cos15° = (√6 + √2)/4

Let’s move toward the next problem. 

• Problem: what is the value of sin 75°?

Solution: 

The angle given in this problem can also be written as 

75° = 45° + 30 °

That means 

Sin75° = sin(45° + 30°)

From the above formulas, we know that.

sin(c+d)=sin(c)cos(d)+cos(c)sin(d)

If we can compare sin(c + d) with sin(45° + 30°), we can find that 

Sin75° = sin45°cos30° + cos45°sin30°

Let’s take the values from the table.

Sin75 =  (1/√2) ⋅ (√3/2) + (1/√2) ⋅ (1/2)

That expression is similar to the above-given problem, so we can directly write the answer.

sin 75° = (√6 + √2)/4

Let’s move towards the next problem

• Problem: what is the value of tan 15°? 

Solution :

Similar to the first question, we can write. 

15° = 60° – 45°

That means 

tan 15° = tan(60°-45°)

From the above formulas, we know that. 

tan(c-d)=tan(c)-tan(d)1+tan(c)tan(d)

If we can compare tan(c – d) with tan(60° + 45°), we can find that. 

tan 15° = (tan 45° –  tan 30°)/(1 + tan 45°tan 30°)

Let’s put the values from the above table:

(tan 45° –  tan30°)/(1 + tan 45°tan30°) = (1 – 1/√3)/(1 + 1 ⋅ 1/√3)

= (√3/√3 – 1/√3)/(√3/√3 + 1/√3)

=[(√3 – 1)/√3]/[(√3 + 1)/√3]

=(√3 – 1)(√3 + 1)

The above-given value is the value of tan 15°.

Conclusion 

In this article, we have discussed the trigonometric ratios for compound angles. We also generated intuition behind the trigonometric ratios and compound angles and found some formulas for trigonometric ratios for compound angles. We also went through some solutions to the problems related to the topic.