Question

Given tanx is 3/4. you can use the double formula to find tan2x. the teacher wants us to find 3 other doubles like sin2x or cos2x without using the double angle formulas.

Trigonometric equation and identitie
ANSWERED
asked 2020-10-28
Given tanx is \(\displaystyle\frac{{3}}{{4}}\). you can use the double formula to find \(\displaystyle{\tan{{2}}}{x}\). the teacher wants us to find 3 other doubles like \(\displaystyle{\sin{{2}}}{x}\) or \(\displaystyle{\cos{{2}}}{x}\) without using the double angle formulas.

Answers (1)

2020-10-29
If \(\displaystyle{\tan{{\left({x}\right)}}}=\frac{{3}}{{4}}\), then \(\displaystyle{\sin{{\left({x}\right)}}}=\frac{{3}}{{5}}{\quad\text{and}\quad}{\cos{{\left({x}\right)}}}=\frac{{4}}{{5}}\), because of Pythagoras’ Theorem.
This is the 3-4-5 right triangle. \(\displaystyle{\tan{{\left({2}{x}\right)}}}={2}\frac{{\tan{{\left({x}\right)}}}}{{{1}-{{\tan}^{{2}}{\left({x}\right)}}}}=\frac{{\frac{{3}}{{2}}}}{{{1}-\frac{{9}}{{16}}}}=\frac{{24}}{{7}}\).
If the legs of a right triangle are 24 and 7, the hypotenuse is \(\displaystyle\sqrt{{{24}^{{2}}+{7}^{{2}}}}=\sqrt{{{576}+{49}}}=\sqrt{{625}}={25}\).
So \(\displaystyle{\sin{{\left({2}{x}\right)}}}=\frac{{24}}{{25}}\) and \(\displaystyle{\cos{{\left({2}{x}\right)}}}=\frac{{7}}{{25}}.\)
\(\displaystyle{\cot{{\left({2}{x}\right)}}}=\frac{{7}}{{24}},{\csc{{\left({2}{x}\right)}}}=\frac{{25}}{{24}},{\sec{{\left({2}{x}\right)}}}=\frac{{25}}{{7}}.\)
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