Cosine theta=8/17 and 270 degrees is < theta, and theta is< 360
θ is in QIV, so x>0 and y<0
cosθ = x/r = 8/17
so,
x = 8
y = -15
r = 17
Now recall the other trig functions:
sinθ = y/r
tanθ = y/x
Sin= -15/17
Cos= 8/17
Tan= -15/8
Csc=-17/15
Sec= 17/8
Cot= -8/15
To find the value of theta, we can use the inverse cosine function (also known as arccosine) since we know the cosine of theta. Let's start step-by-step:
Step 1: Convert the given angle in degrees to radians.
- We are given that 270 degrees < theta < 360 degrees.
- To convert degrees to radians, we use the formula: radians = degrees * (pi/180).
- So, 270 degrees in radians would be: theta_1 = 270 * (pi/180) = (3/2) * pi radians.
Step 2: Use the inverse cosine to find theta.
- We are given that cosine theta = 8/17.
- The inverse cosine (arccosine) of a value between -1 and 1 will give us the associated angle in radians.
- Using the arccosine function, we get: theta_2 = arccos(8/17).
Step 3: Add the two angles.
- As theta lies between 270 degrees and 360 degrees, and we have theta in the second quadrant in radians (theta_1), we add theta_1 and theta_2 to get the final value of theta.
- So, theta = theta_1 + theta_2.
Therefore, the value of theta is theta = (3/2) * pi + arccos(8/17).
To solve this problem, we need to find the value of theta given that the cosine of theta is 8/17 and that 270 degrees is less than theta, and theta is less than 360 degrees.
First, let's find the angle whose cosine is 8/17. We can use the inverse cosine function, also known as arccosine, to do this. The inverse cosine is denoted as cos^(-1) or arccos.
cos^(-1)(8/17) ≈ 0.925
So, we have found that the value of theta whose cosine is 8/17 is approximately 0.925 radians.
Next, we need to check if 270 degrees is less than theta and theta is less than 360 degrees. To convert 270 degrees to radians, we multiply it by (π/180) since there are π radians in 180 degrees.
270 degrees * (π/180) ≈ 4.712 radians
By comparing the value of theta (approximately 0.925) and 4.712, we see that 270 degrees is indeed less than theta. However, theta is less than 360 degrees.
Therefore, the statement "270 degrees is less than theta, and theta is less than 360 degrees" is true for the given value of theta where the cosine of theta is 8/17.