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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​

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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.

User Cesar Hermosillo
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