Question

It is given that cos θ=3/5 and cos ∅= −24/25, where
3π2
sin(θ+∅)


cos (θ−∅)


tan (θ+∅)

Answer 1

Answer

√2 sin∅

Explanation

Given

cos∅ + sin∅ = √2cos∅

Squaring both sides,

(cos∅ + sin∅)² = (√2cos∅)²

⇒ cos²∅ + sin²∅ + 2sin∅cos∅ = 2 cos²∅

⇒ 2sin∅cos∅ = 2cos²∅ - cos²∅ - sin²∅

⇒ 2sin∅cos∅ = cos²∅ - sin²∅

Now, using a² - b² = (a + b)(a - b)

⇒ 2sin∅cos∅ = (cos∅ + sin∅)(cos∅ - sin∅)

Now, we know that (cos∅ + sin∅) = √2cos∅

⇒ 2sin∅cos∅ = (√2cos∅)(cos∅ - sin∅)

⇒ cos∅ - sin∅ = 2sin∅cos∅/√2cos∅

⇒ cos∅ - sin∅ = √2sin∅

Answer 2

Answer:

Firstly, according to your question, [math]sin {\theta}= \dfrac{-3}{5}[/math] and [math]{\pi}< {\theta} < \dfrac{3{\pi}} {2}[/math]

Now, we can consider a right angled triangle say [math]{\Delta}ABC [/math]where angle[math] ABC= {\theta} [/math]

We can ignore the negative sign of [math]sin {\theta}[/math] and find out the value of your desired trigonometric function which is [math]tan {\theta}[/math]

By applying pythagoras’ theorem, the unknown side is 4 units which is the side adjacent to angle [math]{\theta}[/math]

Now we know that [math]tan {\theta}= \dfrac {opposite side}{adjacent side} [/math]

We get [math]tan {\theta}= \dfrac{\pm {3}}{4}[/math]

Here, the value of [math]tan {\theta} [/math]can be [math]{\pm}[/math] depending upon which quadrant it is present in.

Now we can consider the second part of the question, where [math]{\pi}<{\theta} < \dfrac{3{\pi}}{2}[/math]

This just implies that theta lies in the 3rd quadrant, hence

In the third quadrant, tan and cot are postive, you can remember these signs by knowing the acronym All students take chocolates where the initial letters give the positive sign of all functions, sin and its reciprocal, you get the idea…..

Hence, your value of [math]tan {\theta}= \dfrac{+3}{4}[/math]

If you just use the pythagoras theorem and the signs, you can practically get the answer in 2 lines.