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Sine and Cosine

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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Definition of Sine and Cosine

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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.

Definition of Sine and Cosine

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

Sin $\theta$ = $\frac{Opposite}{Hypotenuse}$

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

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


Sine and Cosine Functions

Sin $\theta$ = $\frac{Opposite}{Hypotenuse}$

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

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

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

Properties of Sine and Cosine

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Sine is a periodic and odd function whereas cosine is a periodic and even function.

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.

Sum and Difference Formulas for Sine and Cosine

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

Sine and Cosine Identities:

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

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

Sine and Cosine Table

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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
cos1 $\frac{\sqrt{3}}{2}$$\frac{1}{\sqrt{2}}$$\frac{1}{2}$ 0 -101

Law of Sines and Cosines

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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.

Law of Sines and Cosines

Sine Rule


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


Cosine Rule


a2 = b2 + c2 - 2bc cos A

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

Similarly, the length of b and c.

b2 = a2 + c2 - 2ac cos B

c2 = a2 + b2 - 2ab cos C

Law of Sines and Cosines Problems


For example, let us solve the triangle a = 10, c = 5 and $\theta$ = 45 degree.

Here, the measure of two sides and the angle between them is given.

Law of Sines and Cosines Problems

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

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

=> b2 = a2 + c2 - 2ac cos$\theta$

=> b2 = 102 + 52 - 2 * 10 * 5 cos45o

=> b2 = 100 + 25 - 100 * $\frac{1}{\sqrt{2}}$

= 125 - 70.9

= 54.1

=> b = 7.4 (approx)

Law of Sines and Cosines Word Problems


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

Solved Example

Question: Find the unknown sides and the third angle of the given triangle ABC.

Law of Sines and Cosines Word Problems
Solution:
Given A = 45o, C = 105o 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 = 180o - (A + C)

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

=> B = 48o

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).

Graphing Sine and Cosine Functions

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The sine function and the cosine function have periods of 2$\pi$, therefore, the patterns are repeated to the left and right continuously. The domain of the sine and cosine function is the set of all real numbers, and the range of each function is the interval [-1, 1], and the function has a period of 2$\pi$. The graph of the sine and cosine function is a sin curve and cosine curve respectively.

Sine and Cosine Waves


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

Graphing Sine and Cosine Functions

Sine and Cosine Waves


Amplitude and Period of Sine and Cosine Functions


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 Sine and Cosine

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Hyperbolic trigonometric identities for sine and cosine are given below:

cosh2(x) - sinh2(x) = 1

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

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

Derivative of Sine and Cosine

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Derivative of trigonometric functions, sine and cosine is:

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

$\frac{d}{dx}$
cos x = −sin x

Orthogonality of Sine and Cosine

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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.

Fourier Sine and Cosine Series

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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) = a0 + $\sum_{n = 1}^{\infty}$ an cos nt + $\sum_{n = 1}^{\infty}$ bn sin nt

where,

a0 = $\frac{1}{2\pi}$ $\int_{-\pi}^{\pi}$ f(t) dt

an = $\frac{1}{\pi}$ $\int_{-\pi}^{\pi}$ f(t)cos nt dt, 1$\leq$n

bn = $\frac{1}{\pi}$ $\int_{-\pi}^{\pi}$ f(t)sin nt dt, 1$\leq$n.

Sines and Cosines Problems

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Given below are the examples on sines and cosines.

Solved Examples

Question 1:

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

Sines and Cosines Problems


Solution:

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 300

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



Question 2:

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

Sines and Cosines Problem


Solution:

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



Sine and Cosine Ratios

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We can obtain the sine and cosine ratios from the right angle triangle. In a right angle triangle we have to find the sine ration using the opposite side and hypotenuse. Likewise we have to obtain the cosine rations using the adjacent side and hypotenuse of the right angle triangle. If we have the length of the sides we can find the acute angles of a right angle triangle. And if we know the angles we can find the side lengths using the ratios.

Sum of Angles Formula

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There are several addition formulas in trigonometry sine and cosines. The addition formulas are used to find the ratios when two angles are given. Those addition formulas for sine and cosine are listed below.

Addition Formulas for Sine:

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

Addition Formulas for Cosine:

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

Addition Formulas for Tangent:

tan (A + B) = (tan A - tan B) / (1 - tan A tan B) Based on the sine and cosine we have some formulas
  1. Sin (A + B) = Sin A Cos B + Cos A Sin B
  2. Sin (A - B) = Sin A Cos B - Cos A Sin B
We can write this like the following

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

Likewise, we are having the cosine formula
  1. Cos (A + B) = Cos A Cos B - Sin A Sin B
  2. Cos (A - B) = Cos A Cos B +Sin A Sin B
  3. Cos (A ± B) = Cos A Cos B ± Sin A Sin B
  4. Sin 2A = 2 Sin A Cos A
→ Read More Cosine 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. The Cosine half angle formula is as follows,

$\cos (\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$ → Read More

Sine Half Angle Formula

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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,

$\sin (\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$
More topics in Sine and Cosine
Sine Curve Cosine Curve
Sine and Cosine Ratio Formulas for Sum of Angles
Sine and Cosine Identities Double-angle Formulas
Cosine Half Angle Formula Sine Half-angle Formulas
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