how to i find the six trig values for (7pi/3)
Subtract 2 pi from 7pi/3. The trig functions of 7 pi/3 will be the same values as the trig functions of pi/3, which is 60 degrees.
The cosine of pi/3 is 1/2. That makes the secant 3. The tangent is sqrt3
You should be able to look up or figure out the other funtions of that angle.
Well, let's calculate the six trig values for the angle (7π/3). Just remember, I'm here to have fun with you, not to be a math genius.
First, we need to determine what quadrant (7π/3) is in. Since π is halfway around a circle, 7π/3 is more than 2 full rotations. So, (7π/3) ends up in the third quadrant.
Now, let's start with sine. In the third quadrant, sine is negative. So the sine of (7π/3) is like a pirate treasure - it's not so positive, it's negative! So, sin(7π/3) = -1/2.
Next up is cosine. In the third quadrant, cosine is positive. So, cos(7π/3) = the positive side of deliciousness, which is 1/2.
For tangent, we divide sine by cosine. Tangent is the secret agent of trig values, sneaking around and dividing things. So, tan(7π/3) = (-1/2) ÷ (1/2) = -1.
Now, let's move on to the reciprocal functions. Cosecant is the inverse of sine. So, csc(7π/3) = 1/(-1/2) = -2.
Similarly, secant is the inverse of cosine. Thus, sec(7π/3) = 1/(1/2) = 2.
Lastly, cotangent is the reciprocal of tangent. Therefore, cot(7π/3) = 1/(-1) = -1.
And there you have it! The six trig values for (7π/3) are: sin(7π/3) = -1/2, cos(7π/3) = 1/2, tan(7π/3) = -1, csc(7π/3) = -2, sec(7π/3) = 2, and cot(7π/3) = -1.
To find the six trigonometric values for the angle (7π/3), you can follow these steps:
Step 1: Determine the reference angle:
To find the reference angle, subtract the nearest complete revolution from the given angle. In this case, the nearest complete revolution is 2π.
Reference angle = (7π/3) – (2π) = (π/3).
Step 2: Identify the quadrant:
Since the reference angle (π/3) is positive, we can conclude that the angle (7π/3) is in the fourth quadrant.
Step 3: Find the trigonometric values:
Now, we can determine the trigonometric values using the unit circle or by using trigonometric identities.
Sine (sin):
Since the angle is in the fourth quadrant and the sine function is positive in the second quadrant, the sine value will be positive.
sin(7π/3) = sin(π/3) = √3/2
Cosine (cos):
The cosine function is negative in the fourth quadrant.
cos(7π/3) = cos(π/3) = -1/2
Tangent (tan):
The tangent function is positive in the fourth quadrant.
tan(7π/3) = tan(π/3) = -√3/3
Cosecant (csc):
The cosecant function is reciprocated to the sine function, so it will have the same sign as sine.
csc(7π/3) = csc(π/3) = 2/√3 = 2√3/3
Secant (sec):
The secant function is reciprocated to the cosine function, so it will have the opposite sign.
sec(7π/3) = sec(π/3) = -2
Cotangent (cot):
The cotangent function is reciprocated to the tangent function, so it will have the opposite sign.
cot(7π/3) = cot(π/3) = -√3
Therefore, the six trigonometric values for the angle (7π/3) are as follows:
sin(7π/3) = √3/2
cos(7π/3) = -1/2
tan(7π/3) = -√3/3
csc(7π/3) = 2√3/3
sec(7π/3) = -2
cot(7π/3) = -√3
To find the six trigonometric values for an angle of (7π/3), follow these steps:
Step 1: Convert the angle to radians
(7π/3) is already in radians, so no conversion is needed.
Step 2: Determine the reference angle
The reference angle is the acute angle between the terminal side of the given angle and the x-axis. To find the reference angle, subtract the nearest multiple of π (180 degrees) from the given angle. In this case, the nearest multiple of π that is less than (7π/3) is (6π/3) = 2π. So, the reference angle is 7π/3 - 2π = (π/3).
Step 3: Identify the quadrant
Since the reference angle in Step 2 is (π/3), which is in the first quadrant (between 0 and π/2), the values of the trigonometric functions will be positive.
Step 4: Evaluate trigonometric functions
Using the reference angle, you can now evaluate the trigonometric functions:
- Sine (sin): sin(π/3) = √3/2
- Cosine (cos): cos(π/3) = 1/2
- Tangent (tan): tan(π/3) = (sin(π/3))/(cos(π/3)) = (√3/2)/(1/2) = √3
- Cosecant (csc): csc(π/3) = 1/(sin(π/3)) = 2/√3 = (2√3)/3
- Secant (sec): sec(π/3) = 1/(cos(π/3)) = 2
- Cotangent (cot): cot(π/3) = 1/(tan(π/3)) = 1/√3 = √3/3
Therefore, the six trigonometric values for the angle (7π/3) are:
sin(7π/3) = √3/2
cos(7π/3) = 1/2
tan(7π/3) = √3
csc(7π/3) = (2√3)/3
sec(7π/3) = 2
cot(7π/3) = √3/3