In ∆ABC, ∠A is a right angle, then find the values of sin B, cos C and tan B in each of the following :

a. AB = 12, AC = 5, BC = 13

b. AB = 20, AC = 21, BC = 29

c. BC = √2, AB = AC = 1

Given that ∠A is a right angle.

(a) AB = 12, AC = 5, BC = 13

To Find : sin B, cos C and tan B

We know that,

Here, θ = B

Side opposite to angle B = AC = 5

Hypotenuse = BC =13

So,

Now, Cos C

We know that,

Here, θ = C

Side adjacent to angle C = AC = 5

Hypotenuse = BC =13

So,

Now, tan B

We know that,

Here, θ = B

The side opposite to angle B = AC = 5

The side adjacent to angle B = AB = 12

So,

(b) AB = 20, AC = 21, BC = 29

To Find: sin B, cos C and tan B

We know that,

Here, θ = B

The side opposite to angle B = AC =21

Hypotenuse = BC =29

So,

Now, Cos C

We know that,

Here, θ = C

Side adjacent to angle C = AC = 21

Hypotenuse = BC = 29

So,

Now, tan B

We know that,

Here, θ = B

The side opposite to angle B = AC = 21

The side adjacent to angle B = AB = 20

So,

(c) BC =√2, AB = AC = 1

To Find: sin B, cos C and tan B

We know that,

Here, θ = B

The side opposite to angle B = AC =1

Hypotenuse = BC =√2

So

Now, Cos C

We know that,

Here, θ = C

Side adjacent to angle C = AC = 1

Hypotenuse = BC = √2

So,

Now, tan B

We know that,

Here, θ = B

The side opposite to angle B = AC = 1

The side adjacent to angle B = AB = 1

So,