Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If sin x=(2)/(3),x in quadrant I, then find the exact answers for the following (without finding x ): {:[sin(2x)=],[cos(2x)=],[tan(2x)=]:}

If sin⁡x=23,x \sin x=\frac{2}{3}, x in quadrant I I , then find the exact answers for the following (without finding x x ):\newlinesin⁡(2x)=cos⁡(2x)=tan⁡(2x)= \begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array}

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Find derivatives of sine and cosine functionsright chevron icon

Full solution

Q. If sin⁡x=23,x \sin x=\frac{2}{3}, x in quadrant I I , then find the exact answers for the following (without finding x x ):\newlinesin⁡(2x)=cos⁡(2x)=tan⁡(2x)= \begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array}
  1. Use double angle formulas: Step 11: Use the double angle formulas for sine and cosine.\newlinesin⁡(2x)=2sin⁡(x)cos⁡(x)\sin(2x) = 2\sin(x)\cos(x)\newlinecos⁡(2x)=cos⁡2(x)−sin⁡2(x)\cos(2x) = \cos^2(x) - \sin^2(x)
  2. Calculate cos⁡(x)\cos(x): Step 22: Calculate cos⁡(x)\cos(x) using the Pythagorean identity since sin⁡(x)=23\sin(x) = \frac{2}{3}.cos⁡(x)=1−sin⁡2(x)=1−(23)2=1−49=59=53\cos(x) = \sqrt{1 - \sin^2(x)} = \sqrt{1 - (\frac{2}{3})^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
  3. Substitute into formulas: Step 33: Substitute sin⁡(x)\sin(x) and cos⁡(x)\cos(x) into the double angle formulas.\newlinesin⁡(2x)=2⋅(23)⋅(53)=459\sin(2x) = 2 \cdot \left(\frac{2}{3}\right) \cdot \left(\frac{\sqrt{5}}{3}\right) = \frac{4\sqrt{5}}{9}\newlinecos⁡(2x)=(53)2−(23)2=59−49=19\cos(2x) = \left(\frac{\sqrt{5}}{3}\right)^2 - \left(\frac{2}{3}\right)^2 = \frac{5}{9} - \frac{4}{9} = \frac{1}{9}
  4. Calculate tan⁡(2x)\tan(2x): Step 44: Calculate tan⁡(2x)\tan(2x) using the formula tan⁡(2x)=sin⁡(2x)cos⁡(2x)\tan(2x) = \frac{\sin(2x)}{\cos(2x)}.\newlinetan⁡(2x)=459/19=45\tan(2x) = \frac{4\sqrt{5}}{9} / \frac{1}{9} = 4\sqrt{5}

More problems from Find derivatives of sine and cosine functions

Question
Find the derivative of f(x) f(x) .\newlinef(x)=cos⁡(x) f(x) = \cos(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡(x) f(x) = \tan(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ex f(x) = e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=ln⁡(x) f(x) = \ln(x) \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=tan⁡−1x f(x) = \tan^{-1}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) .\newlinef(x)=xex f(x) = x e^x \newlinef′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=exx f(x) = \frac{e^x}{x} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of f(x) f(x) . \newline f(x)=e(x+1) f(x) = e^{(x + 1)} \newline f′(x)= f'\left(x\right) = ______
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the derivative of f(x) f(x) . \newlinef(x)=x+3 f(x) = \sqrt{x+3} \newlinef′(x)= f'(x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of g(x) g(x). \newline g(x)=ln⁡(2x) g(x) = \ln(2x) \newline g′(x)= g^{\prime}(x) = ______
Get tutor helpright-arrow

Posted 10 months ago

View all (129)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant