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

Finding an angle in a right angle might seen tricky at first, but its's actually quite simple once you know which sides are given. By using basic trigonometric ratios- sine, cosine, and tangent - you can easily determine the unknown angle with just a few calculations.

For example, the figure shown displays a laptop stand where the lengths of two sides are given: 8 cm and 10 cm. You need to find the unknown angle.

Here is how to solve it-

Given sides

• Base = 10 cm
• Height = 8 cm

Using trigonometry ratio we can find angle

tan (θ) =Height / Base
= 8 / 10 = 0.8
θ = tan^-1(0.8) ⩰ 38.66°

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The above example is solved using trigonometric functions which we will be discussing below.

Trigonometric Functions

There are 3 basic trigonometric functions:

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

cosθ = base / hypotenuse

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

sinθ = altitude / hypotenuse

• tanθ: It is the ratio of altitude to 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 the 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 equal to 10cm and hypotenuse equal to 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 the angles of a right-angled 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 + 2 θ = 180^o
3 θ = 180^o - 90^o
3 θ = 90^o
θ = 90^o/3 = 30^o

2 θ = 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 the 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 the hypotenuse, given that 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.

Conclusion

Finding an angle in a right-angled triangle involves using the trigonometric ratios such as the sine, cosine,, or tangent depending on the sides available. By knowing the lengths of two sides, to can apply these ratios to calculate the desired angle. Always remember to use inverse trigonometric functions to find the angle from the ratio of sides. Understanding these fundamental concepts is key to the solving right-angled triangle problems in geometry and trigonometry.