Trigonometry Insights: Area of a Triangle Using Sine

Determine the sides labeled $x$ and $y$ in the triangle if you know that $y$ is $5$ centimeters longer than $x$ and that the area is $6$ square centimeters.

triangle where two sides are x and y cm and they have an included angle of 30 degrees

We know that y is 5 units longer than x. Therefore, we can rewrite y as x+5.

The included angle to the two sides has a measure of 30^(∘). By substituting the angle and the two sides into the formula for calculating a triangle's area using sine, we can create a quadratic equation that contains x.

We can solve this equation by completing the square.

We know that x is 3 centimeters. Then y, which is 5 centimeters longer than x, is 8 centimeters.

Does the triangle for any measure of $∠ v$ have an area of $12$ square centimeters? Please explain your reasoning.

Triangle ABC with one side 4 units long and one side 3 units long.

Let's substitute the given sides and the area we want the triangle to have in the formula for calculating the area of a triangle using sine and try to solve for v.

To calculate the inverse sine of 2 on your graphing calculator, press 2ND, and SIN and then 2. If we try to calculate this value, we notice that we get an error.

The equation sin v=2 can not be solved. This is because the sine value of an acute angle in a right triangle must be less than 1. To understand why, let us have a look at the definition of sine. sin θ =Opposite/Hypotenuse The hypotenuse in a right triangle is always greater than the legs. This means that the sine ratio must be less than 1. Therefore, we can conclude that we cannot have a triangle with sides of 3 and 4 centimeters with an area of 12 square centimeters. That is regardless of the included angle ∠ v at any measure.

The triangle has an area of $3$ square units.

a triangle with two sides of 2.6 units and 4 units and an included angle of v

While sitting in the bus, Kevin wants to determine the measure of $∠ v .$ He makes the following calculation on a piece of paper.

a calculation of an angle using the formula for calculating areas using sine

Kevin looks at his solution, scratches his head, and wonders why the angle looks way bigger than

3 5^{\circ} .

Did Kevin get it right? What angle should

v

be?

If we calculate the inverse sine of 0.5769, we get an angle with a measure of approximately 35^(∘).

However, remember that the sine value is defined as the y-coordinate on the unit circle. There is more than one angle that corresponds to a y-value of 0.5769. Let's view this situation in the unit circle.

Both 35^(∘) and its supplementary angle of 145^(∘) have a sine value of 0.5769. Therefore, the angle Kevin is looking for is 145^(∘), not 35^(∘).

Notice that a graphing calculator always answers with the acute angle of given sine value. If this is not the correct angle, we must calculate the correct one by subtracting the result from 180^(∘).

Sine and the Area of a Triangle

Catch-Up and Review

Investigating the Area of a Triangular Table

Investigating the Area of a Triangle

Area of an Equilateral Triangle

Hint

Solution

Investigating the Area of Obtuse Triangles

Finding the Area of an Obtuse Triangle

Hint

Solution

Area of a Triangle Using Sine

Proof

Using Sine to Calculate Areas in the Real World

Hint

Solution

Practice Using Sine Finding the Area of a Triangle

Finding the Area of a Triangular Table

Hint

Solution

Sine and the Area of a Triangle

Recommended exercises

	10 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions