Sine and Cosine Ratios

Sine and Cosine Values for Common Angles

Here are some sine and cosine values for commonly used angles:

Angle	Sine Value $\sin \theta$	Decimal Value	Cosine Value $\cos \theta$	Decimal Value
$0^\circ$	$0$	$0$	$1$	$1$
$30^\circ$	$\frac{1}{2}$	$0.5$	$\frac{\sqrt{3}}{2}$	$0.87$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$0.71$	$\frac{\sqrt{2}}{2}$	$0.71$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$0.87$	$\frac{1}{2}$	$0.5$
$90^\circ$	$1$	$1$	$0$	$0$

Quantities Found with Sine and Cosine

Sine and cosine are used in the following situations:

Measuring the height of buildings or objects
Navigation and direction finding
Architecture and construction
Physics and engineering
Design and calculation of structures

A Right Triangle Model of a Pyramid

Consider a central cross-section of a regular pyramid through its apex and the midpoint of one base edge. One half of this cross-section is a right triangle. Its vertical leg is the pyramid height, its horizontal leg is half the base width, and its hypotenuse is the slant height from the base-edge midpoint to the apex.

Using Sine to Calculate Pyramid Height

Suppose the slant height is $600 \text{ m}$ and makes an angle of $41^\circ$ with the base. The pyramid height is opposite that angle.

Calculating Pyramid Height with Sine

Right triangle from that cross-section, with the 41 degree angle at the base, the pyramid height opposite it, half the base width adjacent, and the slant height as the hypotenuse.

To calculate the height of the pyramid, we use the sine ratio:

\sin 41^\circ = \frac{\text{pyramid height}}{\text{slant height}}

The modeled pyramid height is approximately $393.6 \text{ m}$ .

Using Cosine to Calculate Half the Base Width

The horizontal leg is adjacent to the $41^\circ$ angle, so cosine gives half the base width:

\cos 41^\circ = \frac{\text{half of base width}}{\text{slant height}}

The full base width in this cross-section is approximately $2y = 905.7 \text{ m}$ .

Differences and Similarities Between Sin Cos and Tan

The three ratios use the same three sides of a right triangle, but each one compares a different pair. Separating the pair a ratio uses from the properties all three share is what keeps sine, cosine, and tangent apart in memory.

Differences

Each line names the pair of sides that one ratio compares, written as a fraction of two sides from the same triangle.

Sine $\sin \theta$ compares the opposite side with the hypotenuse.
Cosine $\cos \theta$ compares the adjacent side with the hypotenuse.
Tangent $\tan \theta$ compares the opposite side with the adjacent side.

Similarities

All three ratios are defined on the same right triangle, so they share several properties. Each one depends only on the angle, which means triangles of different sizes give the same value. The shared properties follow from that common definition, and the division that links all three ratios needs a cosine that does not vanish.

All three are trigonometric ratios in right triangles.
All three change their values according to the angle.
These three ratios have a mathematical relationship:

$\tan \theta = \frac{\sin \theta}{\cos \theta},\qquad \cos \theta \ne 0$

Practice Problem

A child is flying a kite at a height of $3.5 \text{ m}$ above level ground and holds the end of the taut, straight string $60 \text{ cm}$ above the ground. If the string forms an angle of $25^\circ$ with the horizontal, what is its length?

Kite Problem

Visualization of the kite string length problem.

To solve this problem, which trigonometric ratio should we use?

Correct Solution:

We need to calculate the string length (hypotenuse)
We know the effective height of the kite, which is $3.5 \text{ m} - 0.6 \text{ m} = 2.9 \text{ m}$ .
We know the elevation angle $25^\circ$ .
Since we're looking for the hypotenuse and we know the opposite side (effective height), we use the sine ratio:
$\sin 25^\circ = \frac{\text{effective height}}{\text{string length}}$

Rounded to the nearest tenth of a metre, the string is approximately $6.9 \text{ m}$ long.

Sine Ratio

Sine of an angle $\theta$ in a right triangle is the ratio between the length of the opposite side and the hypotenuse.

\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}

Visualization of Sine (

\sin \theta

)

Change the angle in the input below to see how

\sin

changes as the angle changes.

Cosine Ratio

Cosine of an angle $\theta$ in a right triangle is the ratio between the length of the adjacent side and the hypotenuse.

\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}

Visualization of Cosine (

\cos \theta

)