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 A=(9)/(41) and cos B=(21)/(29) and angles A and B are in Quadrant I, find the value of tan(A-B). Answer:

If sin⁡A=941 \sin A=\frac{9}{41} and cos⁡B=2129 \cos B=\frac{21}{29} and angles A and B are in Quadrant I, find the value of tan⁡(A−B) \tan (A-B) .\newlineAnswer:

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Find trigonometric ratios using multiple identitiesright chevron icon

Full solution

Q. If sin⁡A=941 \sin A=\frac{9}{41} and cos⁡B=2129 \cos B=\frac{21}{29} and angles A and B are in Quadrant I, find the value of tan⁡(A−B) \tan (A-B) .\newlineAnswer:
  1. Find cos⁡A\cos A: Use the Pythagorean identity to find cos⁡A\cos A.cos⁡A=1−sin⁡2A\cos A = \sqrt{1 - \sin^2 A}cos⁡A=1−(941)2\cos A = \sqrt{1 - (\frac{9}{41})^2}cos⁡A=1−811681\cos A = \sqrt{1 - \frac{81}{1681}}cos⁡A=16001681\cos A = \sqrt{\frac{1600}{1681}}cos⁡A=4041\cos A = \frac{40}{41}
  2. Find sin⁡B\sin B: Use the Pythagorean identity to find sin⁡B\sin B.
    sin⁡B=1−cos⁡2B\sin B = \sqrt{1 - \cos^2 B}
    sin⁡B=1−(2129)2\sin B = \sqrt{1 - \left(\frac{21}{29}\right)^2}
    sin⁡B=1−441841\sin B = \sqrt{1 - \frac{441}{841}}
    sin⁡B=400841\sin B = \sqrt{\frac{400}{841}}
    sin⁡B=2029\sin B = \frac{20}{29}
  3. Find tan⁡(A−B)\tan(A-B): Use the angle difference identity for tangent to find tan⁡(A−B)\tan(A-B).tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡A⋅tan⁡B\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B}tan⁡A=sin⁡Acos⁡A=9/4140/41=940\tan A = \frac{\sin A}{\cos A} = \frac{9/41}{40/41} = \frac{9}{40}tan⁡B=sin⁡Bcos⁡B=20/2921/29=2021\tan B = \frac{\sin B}{\cos B} = \frac{20/29}{21/29} = \frac{20}{21}tan⁡(A−B)=(940)−(2021)1+(940)⋅(2021)\tan(A-B) = \frac{(\frac{9}{40}) - (\frac{20}{21})}{1 + (\frac{9}{40}) \cdot (\frac{20}{21})}
  4. Simplify tan⁡(A−B)\tan(A-B): Simplify the expression for tan⁡(A−B)\tan(A-B).
    tan⁡(A−B)=(940)−(2021)1+(940)⋅(2021)\tan(A-B) = \frac{(\frac{9}{40}) - (\frac{20}{21})}{1 + (\frac{9}{40}) \cdot (\frac{20}{21})}
    tan⁡(A−B)=(9⋅21)−(20⋅40)40⋅21/(1+9⋅2040⋅21)\tan(A-B) = \frac{(9\cdot21) - (20\cdot40)}{40\cdot21} / (1 + \frac{9\cdot20}{40\cdot21})
    tan⁡(A−B)=189−800840/(1+180840)\tan(A-B) = \frac{189 - 800}{840} / \left(1 + \frac{180}{840}\right)
    tan⁡(A−B)=−611840/(1+180840)\tan(A-B) = \frac{-611}{840} / \left(1 + \frac{180}{840}\right)
    tan⁡(A−B)=−611840/(840+180840)\tan(A-B) = \frac{-611}{840} / \left(\frac{840 + 180}{840}\right)
    tan⁡(A−B)=−611840/(1020840)\tan(A-B) = \frac{-611}{840} / \left(\frac{1020}{840}\right)
    tan⁡(A−B)=−611840⋅8401020\tan(A-B) = \frac{-611}{840} \cdot \frac{840}{1020}
    tan⁡(A−B)=−6111020\tan(A-B) = \frac{-611}{1020}
    tan⁡(A−B)=−6111020\tan(A-B) = \frac{-611}{1020}
  5. Simplify fraction: Simplify the fraction −611/1020-611 / 1020. Both the numerator and the denominator can be divided by 6161. tan⁡(A−B)=−611/1020\tan(A-B) = -611 / 1020 tan⁡(A−B)=−10/17\tan(A-B) = -10 / 17

More problems from Find trigonometric ratios using multiple identities

Question
If cos⁡(θ)=817 \cos(\theta)=\frac{8}{17} and 0∘<θ<90∘ 0^\circ<\theta<90^\circ , what is sec⁡(θ) \sec(\theta) ? Write your answer in simplified, rationalized form. sec⁡(θ)= \sec(\theta)= ______
Get tutor helpright-arrow

Posted 5 months ago

Question
cos⁡(x)=−0.3\cos(x) = -0.3. What is sin⁡(90∘−x)\sin(90^\circ - x)? ____
Get tutor helpright-arrow

Posted 9 months ago

Question
If csc⁡(θ)=178 \csc(\theta) = \frac{17}{8} and 0∘<θ<90∘ 0^\circ < \theta < 90^\circ , what is tan⁡(θ) \tan(\theta) ?\newlineWrite your answer in simplified, rationalized form.\newlinetan⁡(θ)= \tan(\theta) = ______\newline
Get tutor helpright-arrow

Posted 9 months ago

Question
Is the sine function even or odd?\newlineChoices:\newline[A]even\text{[A]even}\newline[B]odd\text{[B]odd}
Get tutor helpright-arrow

Posted 9 months ago

Question
Write the sentence as an equation. \newline8989 is equal to the quotient of yy and 1717 \newlineType a slash ( / / ) if you want to use a division sign. \newline_____
Get tutor helpright-arrow

Posted 7 months ago

Question
Write the sentence as an equation. \newline4545 is equal to the quotient of zz and 55 \newlineType a slash ( / / ) if you want to use a division sign. \newline_____
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for hh. \newline h2+24h=0h^2 + 24h = 0 \newlineWrite each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas. \newline h=h = _____
Get tutor helpright-arrow

Posted 10 months ago

Question
h2−4h=0h^2 - 4h = 0
Get tutor helpright-arrow

Posted 7 months ago

Question
Combine the like terms to create an equivalent expression:\newline−k−(−8k)=□-k-(-8k)=\square
Get tutor helpright-arrow

Posted 10 months ago

Question
Which of the following degree measures is equal to 5π 5 \pi radians?\newline(The number of degrees of arc in a circle is 360360 . The number of radians of arc in a circle is 2π 2 \pi .)\newlineChoose 11 answer:\newline(A) 144∘ 144^{\circ} \newline(B) 900∘ 900^{\circ} \newline(C) 1,080∘ 1,080^{\circ} \newline(D) 1,800∘ 1,800^{\circ}
Get tutor helpright-arrow

Posted 7 months ago

View all (61)

Related topics

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