Question

if tan theta =8 /15,find the value of sin theta +20 /cos theta (1 - cos theta). draw the diagram and show the third side using phytagoras thereom​

Answer 1

Here are the steps to find the value of sin theta + (20 / cos theta) * (1 - cos theta) when tan theta = 8/15:

Step 1: Begin by finding the value of sin theta and cos theta using the given tan theta.

We know that tan theta = opposite/adjacent.

Given tan theta = 8/15, we can assign opposite = 8 and adjacent = 15.

Using the Pythagorean theorem, we can find the hypotenuse as follows:

`\sf\:\text{hypotenuse} = \sqrt{\text{opposite}^2 + \text{adjacent}^2} \\`

`\sf\:= √(8^2 + 15^2) = √(64 + 225) = √(289) = 17 \\`

Therefore, the hypotenuse is 17.

Using the definitions of sine and cosine:

`\sf\:sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = (8)/(17) \\`

`\sf\:cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = (15)/(17) \\`

So, we have sin theta = 8/17 and cos theta = 15/17.

Step 2: Substitute the values of sin theta and cos theta into the given expression:

`\sf\: \text{Expression} = sin(\theta) + (20)/(cos(\theta)) \cdot (1 - cos(\theta)) \\`

Substituting sin theta = 8/17 and cos theta = 15/17:

`\sf\: \text{Expression} = (8)/(17) + (20)/((15)/(17)) \cdot \left(1 - (15)/(17)\right) \\`

Simplifying further:

`\sf\: \text{Expression} = (8)/(17) + (20 \cdot 17)/(15) \cdot \left(1 - (15)/(17)\right) \\`

`\sf\:\text{Expression} = (8)/(17) + (20 \cdot 17)/(15) \cdot \left((17 - 15)/(17)\right) \\`

`\sf\: \text{Expression} = (8)/(17) + (20 \cdot 17)/(15) \cdot \left((2)/(17)\right) \\`

Simplifying further by finding the common denominator:

`\sf\: \text{Expression} = (8)/(17) + (40)/(15) \cdot \left((2)/(17)\right) \\`

`\sf\: \text{Expression} = (8)/(17) + (80)/(15) \cdot \left((2)/(17)\right) \\`

`\sf\: \text{Expression} = (8)/(17) + (16)/(17) \\`

`\sf\: \text{Expression} = (8 + 16)/(17) \\`

`\sf\: \text{Expression} = (24)/(17) \\`

Therefore, the value of the expression is 24/17.