Sine Function Examples

Example 1: Find the value of sin (30°).

Solution:

sin(30°) = sin(π/6 radians)

sin(π/6) is a commonly known value, which is equal to 1/2.

So, sin(30°) = 1/2.

Example 2: Find the value of sin(π/3).

Solution:

sin(π/3) is also a commonly known value, which is equal to √3/2.

Example 3: If sin(x) = 0.8, find the value of x in degrees.

Solution:

To find the value of x, we can take the inverse sine (arcsin) of 0.8:

x = arcsin(0.8)

Using a calculator or table of trigonometric values, you can find that arcsin(0.8) is approximately 53.13°.

Example 4: If sin(2θ) = 0.6 and θ is in the first quadrant, find the value of θ in radians.

Solution:

To find the value of θ, we need to first find the value of 2θ. We can use the inverse sine function:

2θ = arcsin(0.6)

Now, find θ by dividing both sides by 2:

θ = (1/2) × arcsin(0.6)

Using a calculator, you can determine that arcsin(0.6) is approximately 36.87 degrees. So,

θ ≈ (1/2) × 36.87 ≈ 18.44°.

Example 5: Calculate the value of sin(75°) without calculator.

Solution:

We can use the identity sin (x+y) = sin x cos y + sin y cos x

Where x=30° and y=45°

On putting values, we get

sin (75°) = 0.9659

Example 6: If sin(α) = 0.5 and cos(β) = 0.8, find sin(α + β).

Solution:

To find sin(α + β), you can use the sum of angles formula for sine:

sin(α + β) = sin(α) × cos(β) + cos(α) × sin(β)

Using the given values:

⇒ sin(α + β) = 0.5 × 0.8 + √(1 – 0.5²) × √(1 – 0.8²)

⇒ sin(α + β) = 0.4 + √(1 – 0.25) × √(1 – 0.64)

⇒ sin(α + β) = 0.4 + √(0.75) × √(0.36)

⇒ sin(α + β) = 0.4 + 0.866 × 0.6

⇒ sin(α + β) ≈ 0.4 + 0.5196

⇒ sin(α + β) ≈ 0.9196

So, sin(α + β) is approximately 0.9196.

Example 7: An electrician is climbing a 40 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level is 60°.

Solution:

We notice that the figure will be of a right-angled triangle with rope as hypotenuse , vertical pole as perpendicular .

So sin (60°) = Perpendicular/Hypotenuse

Thus, Height of Pole= 0.87 × 40 = 34.8 m [sin (60°) = 0.87]

Sine Function in Trigonometry: Sine Function is one of the six trigonometric ratios which can be defined as the ratio of perpendicular to the hypotenuse in any right angle triangle.

As we know, trigonometry is the branch of mathematics that deals with the relationship between the side and angle of any right-angle triangle. In modern-day, trigonometry is used beyond the right-angle triangle, such as in fields like physics, engineering, computer graphics, and astronomy, and even in everyday applications like GPS navigation and architectural design, so is the Sine Function.

In this article, we will discuss the definition, formula, and various values of the Sine Function, as well as key properties like domain and range, period, and the graph of the Sine Function.