1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ch4aika [34]
3 years ago
7

Find the exact value of the expression. tan( sin−1 (2/3)− cos−1(1/7))

Mathematics
1 answer:
Sonja [21]3 years ago
4 0

Answer:

\tan(a-b)=\frac{2\sqrt{5}-20\sqrt{3}}{5+8\sqrt{15}}

Step-by-step explanation:

I'm going to use the following identity to help with the difference inside the tangent function there:

\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}

Let a=\sin^{-1}(\frac{2}{3}).

With some restriction on a this means:

\sin(a)=\frac{2}{3}

We need to find \tan(a).

\sin^2(a)+\cos^2(a)=1 is a Pythagorean Identity I will use to find the cosine value and then I will use that the tangent function is the ratio of sine to cosine.

(\frac{2}{3})^2+\cos^2(a)=1

\frac{4}{9}+\cos^2(a)=1

Subtract 4/9 on both sides:

\cos^2(a)=\frac{5}{9}

Take the square root of both sides:

\cos(a)=\pm \sqrt{\frac{5}{9}}

\cos(a)=\pm \frac{\sqrt{5}}{3}

The cosine value is positive because a is a number between -\frac{\pi}{2} and \frac{\pi}{2} because that is the restriction on sine inverse.

So we have \cos(a)=\frac{\sqrt{5}}{3}.

This means that \tan(a)=\frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}}.

Multiplying numerator and denominator by 3 gives us:

\tan(a)=\frac{2}{\sqrt{5}}

Rationalizing the denominator by multiplying top and bottom by square root of 5 gives us:

\tan(a)=\frac{2\sqrt{5}}{5}

Let's continue on to letting b=\cos^{-1}(\frac{1}{7}).

Let's go ahead and say what the restrictions on b are.

b is a number in between 0 and \pi.

So anyways b=\cos^{-1}(\frac{1}{7}) implies \cos(b)=\frac{1}{7}.

Let's use the Pythagorean Identity again I mentioned from before to find the sine value of b.

\cos^2(b)+\sin^2(b)=1

(\frac{1}{7})^2+\sin^2(b)=1

\frac{1}{49}+\sin^2(b)=1

Subtract 1/49 on both sides:

\sin^2(b)=\frac{48}{49}

Take the square root of both sides:

\sin(b)=\pm \sqrt{\frac{48}{49}

\sin(b)=\pm \frac{\sqrt{48}}{7}

\sin(b)=\pm \frac{\sqrt{16}\sqrt{3}}{7}

\sin(b)=\pm \frac{4\sqrt{3}}{7}

So since b is a number between 0 and \pi, then sine of this value is positive.

This implies:

\sin(b)=\frac{4\sqrt{3}}{7}

So \tan(b)=\frac{\sin(b)}{\cos(b)}=\frac{\frac{4\sqrt{3}}{7}}{\frac{1}{7}}.

Multiplying both top and bottom by 7 gives:

\frac{4\sqrt{3}}{1}= 4\sqrt{3}.

Let's put everything back into the first mentioned identity.

\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}

\tan(a-b)=\frac{\frac{2\sqrt{5}}{5}-4\sqrt{3}}{1+\frac{2\sqrt{5}}{5}\cdot 4\sqrt{3}}

Let's clear the mini-fractions by multiply top and bottom by the least common multiple of the denominators of these mini-fractions. That is, we are multiplying top and bottom by 5:

\tan(a-b)=\frac{2 \sqrt{5}-20\sqrt{3}}{5+2\sqrt{5}\cdot 4\sqrt{3}}

\tan(a-b)=\frac{2\sqrt{5}-20\sqrt{3}}{5+8\sqrt{15}}

You might be interested in
Px+35=-6x+Q
julia-pushkina [17]
The answer to this question  is C 
6 0
3 years ago
A cone has a base with a radius of 9 mm and a height of 13 mm. What is the volume of the cone?
avanturin [10]

\huge\underline{\red{A}\blue{n}\pink{s}\purple{w}\orange{e}\green{r} -}

  • Given - <u>a </u><u>cone </u><u>with </u><u>base </u><u>radius </u><u>9</u><u>m</u><u>m</u><u> </u><u>and </u><u>height </u><u>1</u><u>3</u><u> </u><u>mm</u>

  • To calculate - <u>volume </u><u>of </u><u>the </u><u>cone</u>

We know that ,

Volume \: of \: cone =  \frac{1}{3}\pi \: r {}^{2}  h \\

<u>su</u><u>b</u><u>stituting </u><u>the </u><u>values</u><u> </u><u>in </u><u>the </u><u>formula</u><u> </u><u>,</u>

Volume =  \frac{1}{3}  \times  3.14  \times 9 \times 9 \times 13 \\  \\  \implies \: 1.05 \times 81 \times 13 \\  \\ \implies \: 1105.65 \: mm {}^{3}

hope helpful ~

7 0
2 years ago
The amount of money you earn varies directly with the number of hours you work. You work for 8 h and earn $98. How many hours do
alexdok [17]
First find the Unit Rate by dividing 98/8 which is : $12.25 per hour. Then divide $441 by $12.25. The answer is 36. You would have to work 36 hours to earn $441.
8 0
3 years ago
Given the function, f(x)=10x-3, <br> what is the value of the function when x=-1/2
Mandarinka [93]
-8

f(-1/2)=10(-1/2)-3
f(-1/2)=(-5)-3
f(-1/2)=-8
6 0
2 years ago
PLZ HELP ME IDK WUT IT IS!
Simora [160]

Answer:

1.  c. 628 cubic cm

2.  c. 502 cubic inches

Step-by-step explanation:

V=\pi r^{2} h\\

1.)  3.14 x 5^2 x 8

3.14 x 25 x 8 = 628

2.)  3.14 x ( 8/2 )^2 x 10

3.14 x 4^2 x 10

3.14 x 16 x 10 = 502.4

5 0
3 years ago
Other questions:
  • PLZ HELP
    11·1 answer
  • The equation of a line given two points needs to be found. Samuel claims that slope-intercept form will generate the equation an
    5·2 answers
  • Construct the perpendicular bisector of the following segments.
    14·1 answer
  • PLEASE HURRY I DONT HAVE MUCH TIME LEFT!
    14·2 answers
  • How to solve In 14 + In x = 0
    7·1 answer
  • Heyyyy I really need help with dis. If u can please answer and explain
    14·2 answers
  • Help! <br><br> . 99 ÷ . 045 =
    11·1 answer
  • 16
    15·1 answer
  • Evaluate the expression. 3.14(7)² =
    7·1 answer
  • Can you please help me ?
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!