Question

If $\sin \theta =\frac{a}{b}$, find sec θ + tan θ in terms of a and b.

Answer 1

Given:

…… (1)

To find:

Now we know, is defined as follows

…… (2)

Now by comparing (1) and (2)

We get,

= a

and

Hypotenuse = b

Therefore triangle representing angle is as shown below

Here side BC is unknown

Now we find side BC by applying Pythagoras theorem to right angled

Therefore,

Now by substituting the value of sides AB and AC from figure (a)

We get,

Therefore,

Now by taking square root on both sides

We get,

Therefore,

Base side …… (3)

Now we know, is defined as follows

Therefore from figure (a) and equation (3)

We get,

…… (4)

Now we know,

Therefore,

Therefore,

…… (5)

Now we know,

Now by substituting the values from equation (1) and (3)

We get,

Therefore,

…… (6)

Now we need to find

Now by substituting the value of and from equation (5) and (6) respectively

We get,

…… (7)

Now we have the following formula which says

Therefore by applying above formula in equation (7)

We get,

Now by substituting in above expression

We get,

Now present in the numerator as well as denominator of above expression gets cancels and we get,

Square root is present in the numerator as well as denominator of above expression

Therefore we can place both numerator as well as denominator under a common square root sign

Therefore,