Trigonometric Comparison: Sin θ and Cos θ

What is Sine Ratio (sin θ)?

Sine of an angle θ 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

)

Move the slider to see how the sine changes as the angle changes.

Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58

30°0.52 Radian

0°360°

What is Cosine Ratio (cos θ)?

Cosine of an angle θ 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

)

Move the slider to see how the cosine changes as the angle changes.

Sin (60°) = 0.87Cos (60°) = 0.50Tan (60°) = 1.73

60°1.05 Radian

0°360°

Sine and Cosine Values for Common Angles

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

Angle	Sine Value (sin θ)	Decimal Value	Cosine Value (cos θ)	Decimal Value
$0°$	$0$	$0$	$1$	$1$
$30°$	$\frac{1}{2}$	$0.5$	$\frac{\sqrt{3}}{2}$	$0.87$
$45°$	$\frac{\sqrt{2}}{2}$	$0.71$	$\frac{\sqrt{2}}{2}$	$0.71$
$60°$	$\frac{\sqrt{3}}{2}$	$0.87$	$\frac{1}{2}$	$0.5$
$90°$	$1$	$1$	$0$	$0$

Applications of Sine and Cosine in Real Life

Sine and cosine have many important applications in everyday life, especially in:

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

Trigonometric Ratios in Pyramids

Let's look at an example of applying sine and cosine in the context of pyramids:

Using Sine to Calculate Pyramid Height

Suppose an archaeologist wants to know the height of a pyramid. They know that the elevation angle from the base to the top of the pyramid is 41° and the slant height (edge) of the pyramid is 600 m.

Calculating Pyramid Height with Sine

Triangle formed when calculating the height of a pyramid.

Sin (41°) = 0.66Cos (41°) = 0.75Tan (41°) = 0.87

41°0.72 Radian

0°360°

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

\sin 41° = \frac{\text{pyramid height}}{\text{slant height}}

\sin 41° = \frac{x \text{ m}}{600 \text{ m}}

0.66 = \frac{x \text{ m}}{600 \text{ m}}

x = 0.66 \times 600 \text{ m} = 396 \text{ m}

Therefore, the height of the pyramid is 396 meters.

Using Cosine to Calculate Pyramid Base Radius

Now, if we want to know the base radius of the pyramid, we can use the cosine ratio:

\cos 41° = \frac{\text{pyramid base radius}}{\text{slant height}}

\cos 41° = \frac{x \text{ m}}{600 \text{ m}}

0.75 = \frac{x \text{ m}}{600 \text{ m}}

x = 0.75 \times 600 \text{ m} = 450 \text{ m}

Therefore, the base radius of the pyramid is 450 meters.

Differences and Similarities Between Sin, Cos, and Tan

Differences

Sine (sin θ) compares the opposite side with the hypotenuse.
Cosine (cos θ) compares the adjacent side with the hypotenuse.
Tangent (tan θ) compares the opposite side with the adjacent side.

Similarities

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

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

A child is flying a kite and has managed to raise it to a height of 3.5 m. The child is holding the string at a height of 60 cm from the ground. If the kite string forms an angle of 25° with the ground, what is the length of the string being used?

Kite Problem

Visualization of the kite string length problem.

Sin (25°) = 0.42Cos (25°) = 0.91Tan (25°) = 0.47

25°0.44 Radian

0°360°

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 (3.5 m - 0.6 m = 2.9 m)
We know the elevation angle (25°)
Since we're looking for the hypotenuse and we know the opposite side (effective height), we use the sine ratio:

$\sin 25° = \frac{\text{effective height}}{\text{string length}}$
$\sin 25° = \frac{2.9 \text{ m}}{x \text{ m}}$
$0.42 = \frac{2.9 \text{ m}}{x \text{ m}}$
$x = \frac{2.9 \text{ m}}{0.42} = 6.9 \text{ m}$

Therefore, the length of the kite string being used is approximately 6.9 meters.

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Trigonometric Comparison: Sin θ and Cos θ