Question

In the given figure, find the value of

1. sin α + cos α
2. tan β − cosec β

Answer 1

1. Finding sin α + cos α  

Step 1: Identify the relevant right-angled triangle and its sides.

The angle α is part of the larger right-angled triangle.

The hypotenuse of this triangle is 41.

The side opposite to α is the vertical line, which has a length of 9.

x² + 9² = 41²    ...[According to Pythagorean theorem]

x² + 81 = 1681

x² = 1681 − 81 = 1600

x = `sqrt1600 = 40`

Step 2: calculating sin α and cos α 

sin α = `"opposite"/"hypotenuse" = 9/41`

cos α = `"opposite"/"hypotenuse" = 40/41`

sin α + cos α    ... [Adding both]

= `9/41 + 40/41`

= `49/41`

= sin α + cos α = `49/41`

Step 3:

2. Finding tan β − cosec β

The angle β is part of the smaller right-angled triangle on the right.
The side opposite to β is 12.

The side adjacent to β is 9.

y² = 9² + 12²       ...[According to Pythagorean theorem]

y² = 81 + 144 = 225 

y = `sqrt225 = 15`

tan β = `"opposit"/"adjacent"`

= `12/9`

= `4/3`

cosec β = `"hypotenuse"/"opposite" = 15/12 = 5/4`

Step 4: Calculating tan β − cosec β

tan β − cosec β = `4/3 − 5/4`

= `4/3 − 5/4`

= `(4 xx 4)/(3 xx 4) − (5 xx 3)/(4 xx 3)`

= `16/12 − 15/12

= `1/12`

tan β − cosec β = `1/12`