1) Find the exact value of the expression:
tan−1(tan(−120651/47π))... How do you find Tan(-120651/47pi)? I don't know how to find exact values, if it's not a recognizable value.
2)Find a simplified expression for tan(sin−1(a/5))...
Because tan = y/x and sin(y/r) can be sin-1(y/r)(i think)
First i did sqrt(5^2-a^2) so i have x
so i did (a^2)/sqrt(5^2-a^2)... I don't know how to do the next step or if i did it right.
3)Solve sin(x)=−0.89 on 0≤x<2π
There are two solutions, A and B, with A < B
1st solution= 4.239
For the first solution all i did was sin-1(-.89)=-1.9073 then i did pi-(-1.9073)to get me 4.2389. And i think that solution is from the 3rd quadrant... and i don't know how to find the other solution...
120651/47 π = 2567.04 π
So, tan(-120651/47 π)
= -tan(120651/47 π)
= -tan(2567π + 2/47π)
= -tan(2/47 π)
arctan(-tan(2/47 π)) = -2/47 π
You are correct. x = a/√(25-a^2)
son tan(arcsin(a/5)) = a/√(25-a^2)
Always use the reference angle from QI.
arcsin(0.89) = 1.097
sin is negative is QIII and QIV
So, your two solutions are π+1.097 and 2π-1.097
1a5. Find tan theta.
1) To find the exact value of tan(-120651/47π), you can first determine the reference angle. The reference angle is the positive acute angle between the terminal side of an angle and the x-axis.
In this case, the angle is -120651/47π, which is negative and larger than -π but smaller than -2π. So, we can find the reference angle by subtracting 2π from -120651/47π until it falls within the range of -π to 0. We get:
-120651/47π - 2π = -120651/47π - (94/47)π = (-120651 - 94)/47π = -120745/47π
Now, we can find the exact value of tan(-120745/47π). The tangent function has a period of π, so we can use the reference angle to find its value in the same quadrant.
Let's say the reference angle is α. We have:
tan(α) = tan(-120745/47π)
To find α, we can solve the equation:
-120745/47π = α + nπ (where n is any integer)
Now we need to determine the quadrant in which -120745/47π falls. Since the angle is negative, it lies in the third quadrant.
To find the exact value of tan(α) in the third quadrant, we need to use the negative value of tan. Hence:
tan(α) = -tan(α)
Therefore, the exact value of tan(-120651/47π) is -tan(α), where α is the reference angle (-120745/47π) in the third quadrant.
2) To find a simplified expression for tan(sin^(-1)(a/5)), let's start by understanding the inverse sine function (sin^(-1)).
sin^(-1)(a/5) represents the angle whose sine is a/5. Let this angle be denoted by α. We have:
sin(α) = a/5
Now, we want to find the value of tan(α). To do this, we can use the Pythagorean identity for sine and cosine:
sin^2(α) + cos^2(α) = 1
Since sin(α) = a/5, we can substitute it into the identity:
(a/5)^2 + cos^2(α) = 1
Simplifying, we get:
a^2/25 + cos^2(α) = 1
cos^2(α) = 1 - a^2/25
cos^2(α) = (25 - a^2)/25
Taking the square root of both sides, we get:
cos(α) = ± √[(25 - a^2)/25]
Now, we can use the definition of tangent:
tan(α) = sin(α)/cos(α)
Substituting the values we found:
tan(α) = (a/5) / [± √((25 - a^2)/25)]
Simplifying, we have:
tan(α) = a / [5 × ± √((25 - a^2)/25)]
Therefore, a simplified expression for tan(sin^(-1)(a/5)) is a / [5 × ± √((25 - a^2)/25)].
3) To solve sin(x) = -0.89 on the interval 0 ≤ x < 2π, you can start by taking the inverse sine (sin^(-1)) of both sides. Keep in mind that the inverse sine function returns the angle whose sine equals the given value.
sin^(-1)(sin(x)) = sin^(-1)(-0.89)
Now, we have:
x = sin^(-1)(-0.89)
Calculating the inverse sine gives us an angle in radians:
x ≈ -1.24167
Since the range specified is 0 to 2π, the value -1.24167 is negative and falls within the fourth quadrant. To find the other solution within the specified range, we can add 2π to the first solution:
x_1 = -1.24167
x_2 = -1.24167 + 2π
Calculating x_2 gives us the second solution:
x_2 ≈ 5.04167
So, the solutions to sin(x) = -0.89 on the interval 0 ≤ x < 2π are approximately x = 4.239 and x = 5.042.