## Class 10 Mathematics Note

# Trigonometry

**Trigonometry **

Area of the triangle =$\frac{1}{2}$base *height

Area of the triangle Δ ABC =$\frac{1}{2}$absinC= $\frac{1}{2}$bcsinA= $\frac{1}{2}$ ca sinB

**Example 1**

Find the area of the triangle Δ ADE

Area of the Δ ADE = $\frac{1}{2}{\rm{\: }}$area of the rectangle ABCD = $\frac{1}{2}{\rm{\: }}$* 8*10 = 40 cm^{2}

**Example 2**

In the given figure , If PQ = 8 cm ,QR = 12cm and ∠ PQR= 30° Find the area of Δ PQR

Soln

Area of the Δ PQR = $\frac{1}{2}$sinQ *PQ *QR

Area of the Δ PQR = $\frac{1}{2}$sin30°*8 * 12

Or, Area of the Δ PQR =$\frac{1}{2}$*$\frac{1}{2}$*8 * 12

= 24 cm^{2}

**Example 3**

Find the length of QT

Area of the Δ PSR = Area of the Δ PQR

Area of the Δ PQR= $\frac{1}{2}{\rm{\: PR\: *QT}}$

16√ 3= $\frac{1}{2}{\rm{\: }}8{\rm{\: *QT}}$

QT =6.9 cm

**Height and Distance **

Angle of elevation: The angle of elevation is defined as angle between the line of sight and horizontal line made by the observer when the observer observes the object above the horizontal line.

Angle of Depression: The angle of depression is defined as angle between the line of sight and horizontal line made by the observer when the observer observes the object below the horizontal line.

**Example 4**

A man observes the top of a pole 52cm height situated in front of him and finds the angle of elevation to be 30° .If the distance between man and pole is 86m .Find the height of that man.

Soln

Tan30° =$\frac{{{\rm{ED}}}}{{{\rm{AD}}}}$=$\frac{{{\rm{ED}}}}{{86}}$

$\frac{1}{{\sqrt 3 }}$=$\frac{{{\rm{ED}}}}{{86}}$

Or, ED = $\frac{{86}}{{\sqrt 3 }}$

Or, ED = 49.65cm

Height of man (DC) = EC – ED = 50- 47.65 =2.35 m

**Example 5**

The angle of elevation from the roof of a house to the top of atop of tree is found to be 30°. If the height of the house and tree are 8 m and 20m respectively, find the distance between the house and the tree.

Soln

ED = EC - DC = 20- 8 = 12 cm

Tan 30°=$\frac{{{\rm{ED}}}}{{{\rm{AD}}}}$

$\frac{1}{{\sqrt 3 }}$=$\frac{{12}}{{{\rm{AD}}}}$

AD =20.78 cm

∴ Distance between tree and house is20.78 cm