**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 90o

**Trigonometric functions**

**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 respect of the lean is adam. 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 respect of the base slant be θ. We can write cosθ = base / hypotenuse = 10/20 = 1/2 θ = cos-1 ( 1/2 ) = 60o

Thus, the value of base angle is 60o.

**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 total of all the three angles in a triangle is 180o. Since one of the angles is 90o and one of the acute angles is doubly the early, we can consider them as θ and 2θ. so, we can write

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90o + θ + 2θ = 180o 3θ = 180o – 90o 3θ = 90o θ = 90o/3 =

30o2θ = 2 × 30o =60oSo, the angles are 30o, 60o,and 90o.

**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 basis distance equals 3m, we can write Hypotenuse = 5m Base = 3m Let the lean of elevation be θ. so, we can write cosθ = Base / Hypotenuse = 3/5 θ = cos-1 ( 3/5 ) θ = 53o thus, the rate of the angle of natural elevation is 53o .

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

Given, the base angle is adequate to 30o and altitude equals 8m, we can apply the sine routine to find the duration of the hypotenuse.

sin30o = altitude / hypotenusehypotenuse = altitude / sin30o Since the measure of sin30o equals 1/2, we can write hypotenuse = altitude / ( 1/2 ) = 2 × altitude thus, hypotenuse = 2 × 8 = 16mThus, the length of the hypotenuse is equal to 16m.

My Personal Notes

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