** **In math, the **sine and cosine** functions are one of the main trigonometric functions. All other trigonometric function can be specified within the expressions of them. The sine and cosine functions are exactly related and can be articulated within conditions of every other. Let us consider A be the angle sine** **can be identified as the ratio of the side opposite near to the angle toward the hypotenuse. Cosine of an angle is the ratio of the side adjacent to the angle A to the hypotenuse.

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Sine and cosine are the important function of the trigonometry. In order to study the sine and cosine, let us assume the right angle triangle the opposed face is opposite to the angle $\theta$. The adjacent side is close to the angle $\theta$. The hypotenuse of triangle is the longest face, which is the one opposed to the right angle.

Let us assume that the sin $\theta$ is the ratio of the side opposite near to the angle toward the hypotenuse.

Again, the cosine of an angle is the ratio of the side adjacent to the angle $\theta$ to the hypotenuse.

Cos $\theta$ = $\frac{Adjacent}{Hypotenuse}$

Cos $\theta$ = $\frac{Adjacent}{Hypotenuse}$

sin$\theta$ = $\frac{1}{\csc \theta}$

cos $\theta$ = $\frac{1}{\sec \theta}$.

The sine is a odd function and periodic with a period of 2$\pi$.

=> sin A = sin(A + 2$\pi$)

and sin (-A) = - sin A

The cosine is a even function and periodic with a period of 2$\pi$.

=> cos A = cos(A + 2$\pi$)

and cos (- A) = cos A.

Below you could see sum and difference formulas for sine and cosine:

cos(A + B) = cos A cos B - sin A sin B

sin(A - B) = sin A cos B - cos A sin B

cos(A - B) = cos A cos B + sin A sin B Given below is the table for the values of sine and cosine

Degrees | 0 |
30 |
45 |
60 |
90 | 180 | 270 |
360 |

Radians | 0 |
$\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\pi$ | $\frac{3\pi}{2}$ | 2$\pi$ |

sin | 0 | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | 1 | 0 | -1 | 0 |

cos | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | 0 | -1 | 0 | 1 |

Law of sines and cosines is also called as the sine and cosine rules. The sin rule and cosine rule allows us to solve problems in triangle which do not contain a right angle. These rules are helpful to find the length of an unknown side and the size of the unknown angle.

$\frac{a}{\sin A}$ = $\frac{b}{\sin B}$ = $\frac{c}{\sin C}$

a

cos A = $\frac{b^2 + c^2 - a^2}{2bc}$

Similarly, the length of b and c.

b

c

=> Let, a = 10, c = 5 and $\theta$ = 45

Apply the law of cosines that involves $\theta$.

=> b

=> b

=> b

= 125 - 70.9

= 54.1

=> b = 7.4 (approx)

Given below are some of the word problems on law of sines and cosines.

Given A = 45^{o}, C = 105^{o} and AB(c) = 16

Find the values of, B, a and b.

**Step 1:**

To find a, use

$\frac{a}{\sin A} = \frac{c}{\sin C}$

=> $\frac{a}{\sin 30} = \frac{16}{\sin 102}$

=> $\frac{a}{\frac{1}{2}} = \frac{16}{0.978}$

=> a = $\frac{32}{0.978}$

=> a = 32.71

**Step 2: **

To find B, Use

B = 180^{o} - (A + C)

The sum of angles A and C is 30^{o }+ 102^{o} = 132^{o}. Since the sum of the angles in a triangle equals 180^{o} we know that angle B must have a measure of 180^{o} - 132^{o} = 48^{o}.

=> B = 48^{o}**Step 3:**

To find b, use

$\frac{b}{\sin B} = \frac{a}{\sin A}$ or $\frac{b}{\sin B} = \frac{c}{\sin C}$

=> $\frac{b}{\sin B} = \frac{c}{\sin C}$

=> $\frac{b}{\sin 48} = \frac{16}{\sin 102}$

=> $\frac{b}{0.74} = \frac{16}{0.978}$

=> b = $\frac{16 * 0.74}{0.978}$

= 12.10 (approx).

Find the values of, B, a and b.

$\frac{a}{\sin A} = \frac{c}{\sin C}$

=> $\frac{a}{\sin 30} = \frac{16}{\sin 102}$

=> $\frac{a}{\frac{1}{2}} = \frac{16}{0.978}$

=> a = $\frac{32}{0.978}$

=> a = 32.71

To find B, Use

B = 180

The sum of angles A and C is 30

=> B = 48

$\frac{b}{\sin B} = \frac{a}{\sin A}$ or $\frac{b}{\sin B} = \frac{c}{\sin C}$

=> $\frac{b}{\sin B} = \frac{c}{\sin C}$

=> $\frac{b}{\sin 48} = \frac{16}{\sin 102}$

=> $\frac{b}{0.74} = \frac{16}{0.978}$

=> b = $\frac{16 * 0.74}{0.978}$

= 12.10 (approx).

The basic graphs of the sine and cosine function is as follows:

The amplitude is one-half the difference between the maximum and minimum values for a periodic function. The amplitude and period of sine and cosine functions involves the distance from the midpoint to the highest or lowest point of the function and the distance between any two repeating points on the function.

For the curves, y = sin x and y = cos x

Amplitude of Sine and Cosine Functions = $\frac{1 - (-1) }{2}$ = 1

Period of Sine and Cosine Functions = 2$\pi$.

Hyperbolic trigonometric identities for sine and cosine are given below:

cosh

sinh(x) = $\frac{1}{2}$$(e^x - e^{-x})$

cosh(x) = $\frac{1}{2}$$(e^x + e^{-x})$

Derivative of trigonometric functions, sine and cosine is:

$\frac{d}{dx}$ sin x = cos x

$\frac{d}{dx}$ cos x = −sin x

Two non-zero functions f and g are said to be orthogonal on a $\leq$ x $\leq$ b if,

$\int_{a}^{b}$ f.g dx = 0

To check whether functions are orthogonal or not, multiply both the functions and then integrate the resultantant curve, obtaining a sum of zero. A sine and cosine are orthogonal to each other, if

$\int_{0}^{2\pi}$ sin x cos x = 0.

A fourier series is essentially a means of expressing any periodic function as a sum of sines or cosines of different frequencies. A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.

f(t) = a

where,

a

a

b

Given below are the examples on sines and cosines.

Find the opposite side of the right angle triangle, if the angle $\theta$ is 30 and the hypotenuse side is 33.

Given, $\theta$ = 30 degree and the hypotenuse side of the triangle is 33

Let 'a' be the length of the opposite side of the right angle triangle.

We know that, Sin $\theta$ = $\frac{Opposite}{Hypotenuse}$=> Sin 300 = $\frac{a}{33}$

a = 33 x Sin 30^{0}

a = 33 x 0.5 {Value of sin 30 degree is 0.5}

a = 16.5

Thus, the opposite side of the triangle = 16.5

Find the hypotenuse side of the right angle triangle if angle $\theta$ is 60Â°**,** and the adjacent side is 14.

Given, $\theta$ = 60 degree and the adjacent side of the triangle is 14.

Let 'a' be the length of the hypotenuse of the right angle triangle.

Cos $\theta$ = $\frac{Adjacent}{Hypotenuse}$Cos 60 = $\frac{14}{a}$

a = $\frac{14}{0.5}$

a = $\frac{14}{0.5}$

a = 28

Thus, the hypotenuse side of the triangle = 28

sin (A + B) = sin A cos B + cos A sin B

cos (A + B) = cos A cos B - sin A sin B

tan (A + B) = (tan A - tan B) / (1 - tan A tan B) Based on the sine and cosine we have some formulas

- Sin (A + B) = Sin A Cos B + Cos A Sin B
- Sin (A - B) = Sin A Cos B - Cos A Sin B

Sin (A ± B) = Sin A Cos B ± Cos A Sin B

Likewise, we are having the cosine formula

- Cos (A + B) = Cos A Cos B - Sin A Sin B
- Cos (A - B) = Cos A Cos B +Sin A Sin B
- Cos (A ± B) = Cos A Cos B ± Sin A Sin B
- Sin 2A = 2 Sin A Cos A

$\cos (\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$ → Read More Sine half angle formula for trigonometric functions are derived from the sum of angles formula. Sine, Cosine and Tangent are the trigonometric functions involved in half angle formulas. Sine half angle formula is as follows,

$\sin (\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

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