Question

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If tan $\large\theta$ = $\large \frac{a}{b}$ , then cos $\large\theta$ =

a, $\Large \frac{a}{a² + b²}$

b, $\Large \frac{b}{a² + b²}$

c, $\Large \frac{a}{ \sqrt{a² + b²} }$

d, $\Large \frac{b}{ \sqrt{a² + b²} }$ .

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Answer 1

Required Answer:-

Whenever there is such type of questions, always try to establish a relation between the trigonometric ratios like here tan and cos. Let's look at some identities that relates them.

• sec²θ - tan²θ = 1
• cos θ = 1/sec θ

Given,

tan θ = a/b

Squaring both sides,

tan²θ = a²/b²

Then, according to identity sec²θ - tan²θ = 1,

sec²θ - a²/b² = 1

sec²θ = 1 + a²/b²

sec²θ = (a² + b²)/b²

Now square root both sides,

sec θ = √(a² + b²)/b

Taking reciprocal because of sec and cos relation,

cos θ = b/√(a² + b²)

That's Option (D) b/√(a² + b²)

Note:-

• Practice trigonometric identities and reciprocal relations between the ratios.
• Have a grip on them first, try to prove them using a simple triangle angle-side relatio$.$

Answer 2

Given : If tan θ = a/b, then cos θ = ?

Options are :

• a/a² + b²

• b/a² + b²

• √a/a² + b²

• √b/a² + b²

Need to Find : The value of cos θ

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★ Cᴏɴᴄᴇᴘᴛ :

According to the question, here we are given that tan θ is a/b and we beed to get cos θ. So, for finding it first we need to get sec θ using a formula then after which we know that cos θ is the reciprocal of sec θ hence by reciprocating it we can get the required answer

★ Sᴏʟᴜᴛɪᴏɴ :

• tan θ = a/b

So, now as we know that we need to get sec θ. Therefore using

❍ sec² θ - tan² θ = 1 ❍

Putting the values in the above formula

• sec² θ - (a/b)² = 1

• sec² θ = 1 + a²/b²

Now, getting the LCM

• sec² θ = b²/b² + a²/b²

• sec² θ = b² + a²/b²

Taking sq. root on the other side

• sec θ = √(b² + a²)/b²

As we know that reciprocal of sec θ is cos θ. So,

• cos θ = 1/sec θ

• cos θ = √b²/a² + b²

• cos θ = b/√a² + b²

␥ Henceforth, the answer is Option D i.e. b/√a² + b²