How to find an Angle in a Right-Angled Triangle?

Triangles are three-sided closed polygons formed by the intersection of three lines. It is encountered a lot in everyday life. It is one of the basic shapes of geometry. It has three sides, three angles, and three vertices. A Right Angled Triangle is one where one of the angles is always equal to 90°. Pythagoras Theorem is derived for Right-angled triangles, Which states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of base and perpendicular.

Given the length of at least two sides of a Right-Angled triangle, we can find the value of any angle of the right-angled triangle. For this, we use various trigonometric functions like sine, cosine, tangent, cotangent, sec, and cosec. These help us to relate the angles of a right-angled triangle with its sides.

Properties

• There is a right-angle vertex among the three vertices
• The side opposite to the right-angled vertex is called the hypotenuse.
• The length of the sides follows the Pythagoras theorem, which states

hypotenuse² = base² + altitude²

• The hypotenuse is the longest side of a right-angled triangle.
• The angles other than the right angle are acute angles since the value is less than 90^o

Trigonometric functions

ABC is a right-angled triangle with ∠B as right-angle

• cosθ: This gives the ratio of the base by the hypotenuse of a right-angled triangle.

cosθ = base / hypotenuse

• sinθ: This gives the ratio of altitude by the hypotenuse of a right-angled triangle.

sinθ = altitude / hypotenuse

• tanθ: It is the ratio of altitude by the base of a right-angled triangle.

tanθ = altitude / base

• cotθ: It is the inverse of tanθ
• secθ: It is the inverse of cosθ
• cosecθ: It is the inverse of sinθ

To find the angles of a right-angled triangle, we can take the trigonometric inverse of the ratio of given sides of the triangle.

Example:

If sinθ = x, then we can write

θ = sin^-1x.

This returns the angle for which the sine value of the angle is x.

Similarly, there exists cos^-1θ, tan^-1θ, cot^-1θ, sec^-1θ, and cosec^-1θ

Sample Problems

Question 1. Given a right-angled triangle, with base equals 10cm and hypotenuse equals 20cm. Find the value of the base angle.

Solution:

Given, Base = 10cm

Hypotenuse = 20cm

Let, the value of the base angle be θ. We can write

cosθ = base / hypotenuse = 10/20 = 1/2

θ = cos^-1(1/2) = 60^o

Thus, the value of base angle is 60^o.

Question 2. Find the value of angles of a right angles triangle, given that one of the acute angles is twice the other.

Solution:

Since we know the sum of all the three angles in a triangle is 180^o.

Since one of the angles is 90^o and one of the acute angles is twice the other, we can consider them as θ and 2θ.

So, we can write

90^o + θ + 2θ = 180^o

3θ = 180^o – 90^o

3θ = 90^o

θ = 90^o/3 = 30^o

2θ = 2 × 30^o = 60^o

So, the angles are 30^o, 60^o, and 90^o.

Question 3. Find the value of the angle of elevation of a ladder of length 5m, given that base of the ladder is at a distance of 3m from the wall.

Solution:

Since the ladder acts as a hypotenuse of a right angles triangle and base distance equals 3m, we can write

Hypotenuse = 5m

Base = 3m

Let the angle of elevation be θ. So, we can write

cosθ = Base / Hypotenuse = 3/5

θ = cos^-1(3/5)

θ = 53^o

Thus, the value of the angle of elevation is 53^o.

Question 4. Find the value of hypotenuse, given the length of the altitude is 8m and the base angle equals 30^o.

Solution:

Given, the base angle is equal to 30^o and altitude equals 8m, we can apply the sine function to find the length of the hypotenuse.

sin30^o = altitude / hypotenuse

hypotenuse = altitude / sin30^o

Since the value of sin30^o equals 1/2, we can write

hypotenuse = altitude / (1/2) = 2 × altitude

Thus, hypotenuse = 2 × 8 = 16m

Thus, the length of the hypotenuse is equal to 16m.