Without using a calculator or mathematical table find the value of x in sin 60°=cos (3x-45°)0°<x<π\2
Since sin 60° = 1/2, we have
1/2 = cos(3x - 45°)
Taking the cosine of both sides, we get
cos(90° - 60°) = cos(3x - 45°)
cos(30°) = cos(3x - 45°)
To solve for x, we need to find the values of 3x - 45° that have a cosine of 1/2. Since the cosine function has a period of 360°, we can write
cos(3x - 45°) = cos(360° - (3x - 45°))
= cos(405° - 3x)
Now we need to find the values of 405° - 3x that have a cosine of 1/2. These values occur when
405° - 3x = ±60° + 360°n
where n is an integer. Solving for x, we get
x = (405° ± 60° - 360°n) / 3
= (135° ± 20° - 120°n) / 1
= 135°/3 ± 20°/3 - 120°n/3
= 45° ± (4/3)° - 40°n
Note that the condition 0° < x < π/2 implies that n = 0, since otherwise x would be outside the given range. Therefore, we have
x = 45° + (4/3)°
or
x = 45° - (4/3)°
These values can be simplified as
x ≈ 45.333°
or
x ≈ 44.667°
Therefore, the value of x is approximately 45.333° or 44.667°.
Given that sin x=2/3 and x is acute angle find tan x
We know that sin x = opposite/hypotenuse, so we can draw a right triangle with opposite side 2 and hypotenuse 3. Using the Pythagorean theorem, we can find the adjacent side:
a^2 + b^2 = c^2
a^2 + 2^2 = 3^2
a^2 + 4 = 9
a^2 = 5
a = √5
Therefore, tan x = opposite/adjacent = 2/√5. To rationalize the denominator, we can multiply by √5/√5:
tan x = 2/√5 * √5/√5
tan x = 2√5/5
So the value of tan x is 2√5/5.
Given that sin x = 2/3 is an acute angle find tan x
We can start by drawing a right triangle with angle x as shown in the figure below:
```
|
|
2 | /\
---- | / \
3 | /_____\
x |<-2-><-√5->
```
Here, sin x = 2/3, which means that the opposite side is 2 and the hypotenuse is 3. We can use the Pythagorean theorem to find the third side:
a^2 + b^2 = c^2
a^2 + 2^2 = 3^2
a^2 = 3^2 - 2^2
a^2 = 5
a = √5
Therefore, tan x = opposite/adjacent = 2/√5. To rationalize the denominator, we can multiply both numerator and denominator by √5:
tan x = 2/√5 * √5/√5
tan x = 2√5/5
So the value of tan x is 2√5/5.
since sin(60) = cos(30), we have
cos(30) = cos(3x-45)
3x-45 = 30
3x = 75
x = 25
I'm sorry, but this solution is incorrect.
First, we need to recognize that cos(30) = sin(60) and not cos(60), as stated in the problem.
Secondly, when we set cos(30) = cos(3x - 45), we cannot immediately equate the angles 30 and 3x - 45.
This is because the cosine function is not one-to-one; it has a period of 360 degrees. Thus, we need to add or subtract multiples of 360 degrees to the angle on one side of the equation to find all possible solutions.
So, starting from cos(30) = cos(3x - 45), we can add 360 degrees to either side to get:
cos(30) = cos(3x - 45) + 360
We can also subtract 360 degrees from either side to get:
cos(30) = cos(3x - 45) - 360
To solve for x, we need to first find all possible values of 3x - 45 that have a cosine of cos(30) = √3/2.
Using the formula cos(θ) = adj/hyp, we consider a right triangle where the adjacent side is √3 and the hypotenuse is 2. Therefore, the corresponding acute angle is 30 degrees.
Now, we look for angles in degrees that are co-terminal with 30 degrees (either in the same quadrants or in the corresponding quadrants), and that also satisfy:
cos(3x - 45) = √3/2
One such angle is 75 degrees.
Thus, we have:
3x - 45 = 75
Solving for x, we get:
3x = 120
x = 40 degrees
Therefore, the solution is x = 40 degrees, not 25.