Subscribe to our YouTube channel for the latest videos, updates, and tips.


Proof of Compound Angle Formula
sin (α + β)

We will learn step-by-step the proof of compound angle formula sin (α + β). Here we will derive formula for trigonometric function of the sum of two real numbers or angles and their related result. The basic results are called trigonometric identities.

The expansion of sin (α + β) is generally called addition formulae. In the geometrical proof of the addition formulae we are assuming that α, β and (α + β) are positive acute angles. But these formulae are true for any positive or negative values of α and β.

Now we will prove that, sin (α + β) = sin α cos β + cos α sin β; where α and β are positive acute angles and α + β < 90°.

Let a rotating line OX rotate about O in the anti-clockwise direction. From starting position to its initial position OX makes out an acute ∠XOY = α.

Again, the rotating line rotates further in the same direction and starting from the position OY makes out an acute ∠YOZ = β.

Thus, ∠XOZ = α + β < 90°.    

We are suppose to prove that, sin (α + β) = sin α cos β + cos α sin β.



Construction: On the bounding line of the compound angle (α + β) take a point A on OZ, and draw AB and AC perpendiculars to OX and OY respectively. Again, from C draw perpendiculars CD and CE upon OX and AB respectively.

Proof of Compound Angle Formula sin (α + β)

Proof: From triangle ACE we get, ∠EAC = 90° - ∠ACE = ∠ECO = alternate ∠COX = α.

Now, from the right-angled triangle AOB we get,

sin (α + β) = ABOA

               = AE+EBOA

               = AEOA + EBOA

               = AEOA + CDOA

               = AEACACOA + CDOCOCOA

               = cos ∠EAC sin β + sin α cos β

               = sin α cos β + cos α sin β, (since we know, ∠EAC = α)

Therefore, sin (α + β) = sin α cos β + cos α sin β.         Proved.


1. Using the t-ratios of 30° and 45°, evaluate sin 75°

Solution:

sin 75°

= sin (45° + 30°)

= sin 45° cos 30° + cos 45° sin 30

= 1232 + 1212

= 3+122

 

2. From the formula of sin (α + β) deduce the formulae of cos (α + β) and cos (α - β).

Solution:

We know that, sin (α + β) = sin α cos β + cos α sin β …….. (i)

Replacing α by (90° + α) on both sides of (i) we get,

sin (90° + α + β)

= sin {(90° + α) + β} = sin (90° + α) cos β + cos (90° + α) sin β, [Applying the formula of sin (α + β)]

⇒ sin {90° + (α + β)} = cos α cos β - sin α sin β, [since sin (90° + α) = cos α and cos (90° + α) = - sin α]

⇒ cos (α + β) = cos α cos β - sin α sin β …….. (ii)

Again, replacing β by (- β) on both sides of (ii) we get,

cos (α - β) = cos α cos (- β) - sin α sin (- β)

⇒ cos (α - β) = cos α cos β + sin α sin β, [since cos (- β) = cos β and sin (- β) = - sin β]

 

3. If sin x = 35, cos y = -1213 and x, y both lie in the second quadrant, find the value of sin (x + y).

Solution:

Given, sin x = 35, cos y = -1213 and x, y both lie in the second quadrant.

We know that cos2 x = 1 - sin2 x = 1 - (35)2 = 1 - 925 = 1625

⇒ cos x = ± 45.

Since x lies in the second quadrant, cos x is – ve

Therefore, cos x = -45.

Also, sin2 y = 1 - cos2 y = 1 - (-1213)2 = 1 - 144169 = 25169

⇒ sin y = ± 513

Since y lies in the second quadrant, sin y is + ve

Therefore, sin y = 513

Now, sin (x + y) = sin x cos y + cos x sin y

                       = 35 ∙ (- 1213) + (- 45) ∙ 513

                       = - 3665 - 2065

                       = - 5665


4. If m sin (α + x) = n sin (α + y), show that, tan α = nsinymsinxmcosxncosy

Solution:

Given, m sin (α + x) = n sin (α + y)

Therefore, m (sin α cos x + cos α sin x) = n (sin α cos y+ cos α sin y), [Applying the formula of sin (α + β)]

m sin α cos x + m cos α sin x = n sin α cos y + n cos α sin y,

or, m sin α cos x - n sin α cos y = n cos α sin y - m cos α sin x

or, sin α (m cos x - n cos y) = cos α (n sin y - m sin x)

or, sinαcosαnsinymsinxmcosxncosy.           

or, tan α = nsinymsinxmcosxncosy.   Proved.

 Compound Angle





11 and 12 Grade Math

From Proof of Compound Angle Formula sin (α + β) to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  2. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More