Question

practice assignment 9.3 double-angle and half-angle identities
score: 5/6 answered: 5/6
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question 6
if \\( \tan ( x ) = \frac { 21 } { 20 } \\) (in quadrant-i), find
\\( \sin ( 2 x ) = \\)
(please enter answer accurate to 4 decimal places.)

Explanation:

Step1: Find \(\sin x\) and \(\cos x\)

Given \(\tan x=\frac{21}{20}=\frac{\sin x}{\cos x}\), and \(\sin^{2}x+\cos^{2}x = 1\).
Since \(\tan x=\frac{21}{20}\), let \(\sin x = 21k\) and \(\cos x=20k\).
Substitute into \(\sin^{2}x+\cos^{2}x = 1\): \((21k)^{2}+(20k)^{2}=1\), \(441k^{2}+400k^{2}=1\), \(841k^{2}=1\), \(k^{2}=\frac{1}{841}\), \(k=\frac{1}{29}\) (because \(x\) is in Quadrant - I, \(k>0\)).
So \(\sin x=\frac{21}{29}\), \(\cos x=\frac{20}{29}\).

Step2: Use double - angle formula

The double - angle formula for sine is \(\sin(2x)=2\sin x\cos x\).
Substitute \(\sin x=\frac{21}{29}\) and \(\cos x=\frac{20}{29}\) into the formula:
\(\sin(2x)=2\times\frac{21}{29}\times\frac{20}{29}=\frac{840}{841}\approx0.9988\)

Answer:

\(0.9988\)