tan(x)=5 sin(x) for interval -π < x < π
A) 0, 1.571 B) -1.571, 0, 1.571 C) -1.369, 0, 1.369 D) 0, 1.369
recall that tan(x) can be rewritten as
tan (x) = sin (x) / cos (x)
substituting:
sin(x) / cos(x) = 5 sin(x)
the sin(x) will be cancelled:
1/cos(x) = 5
cos(x) = 1/5
solving this,
x = +/- 1.369
since it must be on interval -π < x < π
x = - 1.369
are we solving ????
tanx = 5sinx
sinx/cosx= 5sinx
sinx = 5sinxcosx
sinx - 5sinxcos)=0
sinx(1 - 5cosx) = 0
sinx = 0 or cosx = 1/5
if sinx = 0, x = 0, π or 2π
if cosx = 1/5, x = 1.369 or -1.369 if -π < x < π
so for the given domain
x = -1.369 , 0, 1.369 , which would be choice C)
Thanks so much guys!!!!
Did you notice that Jai missed one of the answers of
x = 0.
You should not cancel sinx , but rather use it as one of the factors.
by canceling sinx , he "lost" the answer to sinx = 0
oh yeah,, sorry about that. 0 is also a solution~
thanks for correcting me, sir~ :)
To solve the equation tan(x) = 5 sin(x) for the given interval -π < x < π, you need to find the values of x where the equation is true.
Step 1: Rearrange the equation
Start by rearranging the equation to isolate one of the trigonometric functions. In this case, we will isolate the tan(x):
tan(x) = 5 sin(x)
Divide both sides by sin(x):
tan(x) / sin(x) = 5
Step 2: Apply the identity
Next, use the identity that tan(x) = sin(x) / cos(x):
(sin(x) / cos(x)) / sin(x) = 5
Simplify by canceling out the sin(x):
1 / cos(x) = 5
Step 3: Solve for cos(x)
Now, solve for cos(x) by taking the reciprocal of both sides:
cos(x) = 1 / 5
Step 4: Find the angle
Use the inverse cosine function (also known as arccos) to find the angle:
x = arccos(1 / 5)
Step 5: Evaluate the angle
Evaluate the arccosine using a calculator or table of trigonometric values. The interval given is -π < x < π, so the answer should fall within this range.
Using the calculator, you will find that arccos(1/5) ≈ 1.369
Therefore, the answer is option D) 0, 1.369.