Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application
Cos (x + y) × cos y + sin (x + y) × sin y = cos x
Firstly, we will be using the trigonometric identities.
As we know that:
Cos (a+b) = cosa × cosb – sinxa × sinb
Sin (a+b) = sina × cosb + sina × cosb
Now, we will be substituting these formulas into the equation.
⟹ [cosx × cosy – sinx × siny) × cosy + (sinx × cosy + cosx × siny) × siny
Now expand the brackets
⟹ cosx × cos²y – sinx × siny × cosy + sinx × siny × cosy + cosx × sin²y
Further, reduce the similar terms,
⟹ cosx × cos²y + cosx × sin²y
Now, factor Cos x from both the terms.
⟹ cos x × (cos²y + sin²y)
As, sin^2 y + cos^2 y = 1
⟹ cosx × 1 = cosx
The functions of sine and cosine are periodic having “2p” period. This signifies that
Sin (q) = Sin (q + 2p)
Cos (q) = Cos (q + 2p)
Or normally,
Sin (q) = Sin (q + 2pk)
Cos (q) = Cos (q + 2pk),
Where, k Î integers.
As Sine is an odd function; thus,
sin(-q) = -sin(q)
And Cosine is an even function; thus,
cos(-q) = cos(q)
Formula says:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
It is then easily derived that
sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
Or more generally,
sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)
cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
It is then easily derived that
cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
or,
cos (x – y) = cos (x) cos (y) + sin(x)sin(y)
Or more generally,
Cos (a ± b) = cos (a) cos (b) (-/+) sin (a) sin (b)
or,
cos (x ± y) = cos (x) cos (y) (-/+) sin (x) sin (y)
From the sine equation above, we can obtain that
Sin (2x) = 2sin(x) cos(x)
From the cosine equation above, we can obtain that
Cos (2x) = cos2(x) – sin2(x)
Simple Applications of Trigonometry
Trigonometry has a vast range of applications. Its various fields include:
- Development in computer music: Have you heard of sound travels in a wave pattern and this waveform via Sine or Cosine function for the computer music development.
- Using trigonometric functions, you can measure the height of a mountain or a building. The distance can also be measured from the elevation angle and the viewpoint.
- Trigonometry is even used in Aviation: To measure direction, distance, and speed of the wind, trigonometry is used.
- Used in Criminology: In the investigation of the crime happening around the world, Trigonometry has a huge role to play. Trigonometric functions are significant and beneficial in calculating projectile’s trajectory; also helps in calculating the collision causes in a car accident. It also identifies that at what angle the gun is being shot & how the object falls.
- Use in Sea Biology: The trigonometry is oftentimes used by the biologists of marine for the measurement of figuring out the sunlight’s dept which affects algae for the photosynthesis process. The use of the trigonometry is also helpful in measuring or estimating animals’ size such as whales and also helpful in understanding the behaviour of the marine animals.
- Advantageous in Navigation: Trigonometry is helpful in the navigation of directions. It estimates that in what direction you should place the compass for getting the straight direction. Also helps in pinpointing the location and even finding the distance along with seeing the horizon.
- The calculus is based on the algebra and trigonometry
- The basic trigonometric functions such as sine & cosine are used for describing light and sound waves.
- Trigonometry is even used in oceanography for calculating waves’ heights and tides in the oceans.
- Helpful in map creation
- Useful in satellite systems
