Math

Question Find tanhx\tanh x given sinhx=34\sinh x = -\frac{3}{4}.

Studdy Solution

STEP 1

Assumptions
1. The given value is sinhx=34\sinh x = \frac{-3}{4}.
2. We need to find the value of tanhx\tanh x.
3. We know that tanhx=sinhxcoshx\tanh x = \frac{\sinh x}{\cosh x}.
4. We also know the identity cosh2xsinh2x=1\cosh^2 x - \sinh^2 x = 1.

STEP 2

First, we will use the identity cosh2xsinh2x=1\cosh^2 x - \sinh^2 x = 1 to find coshx\cosh x.
cosh2x(34)2=1\cosh^2 x - \left(\frac{-3}{4}\right)^2 = 1

STEP 3

Calculate the square of sinhx\sinh x.
(34)2=916\left(\frac{-3}{4}\right)^2 = \frac{9}{16}

STEP 4

Substitute the value of sinh2x\sinh^2 x into the identity.
cosh2x916=1\cosh^2 x - \frac{9}{16} = 1

STEP 5

Add 916\frac{9}{16} to both sides of the equation to isolate cosh2x\cosh^2 x.
cosh2x=1+916\cosh^2 x = 1 + \frac{9}{16}

STEP 6

Calculate the right side of the equation.
cosh2x=1616+916\cosh^2 x = \frac{16}{16} + \frac{9}{16}
cosh2x=2516\cosh^2 x = \frac{25}{16}

STEP 7

Take the square root of both sides to find coshx\cosh x. Note that coshx\cosh x is always positive, so we only consider the positive square root.
coshx=2516\cosh x = \sqrt{\frac{25}{16}}

STEP 8

Calculate the square root.
coshx=54\cosh x = \frac{5}{4}

STEP 9

Now that we have both sinhx\sinh x and coshx\cosh x, we can find tanhx\tanh x using the definition tanhx=sinhxcoshx\tanh x = \frac{\sinh x}{\cosh x}.

STEP 10

Substitute the values of sinhx\sinh x and coshx\cosh x into the definition of tanhx\tanh x.
tanhx=3454\tanh x = \frac{\frac{-3}{4}}{\frac{5}{4}}

STEP 11

Divide the fractions.
tanhx=3445\tanh x = \frac{-3}{4} \cdot \frac{4}{5}

STEP 12

Multiply the numerators and the denominators.
tanhx=3445\tanh x = \frac{-3 \cdot 4}{4 \cdot 5}

STEP 13

Simplify the fraction by canceling out the common factor of 4.
tanhx=35\tanh x = \frac{-3}{5}
The value of tanhx\tanh x is 35\frac{-3}{5}.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord