If cos 0 = - 1/2 and tan 0 > 0, find the quadrant that contains the terminal side of 0, and then find the exact values of the other five trig functions of 0.
The cosine is positive in I and IV
quadrant,negative in II and III quadrant.
The tangent is positive in I and III guadrant,negative in II and IV quadrant.
If cos(theta)<0 cosine is negative
If tan(theta)>0 tangent is positive
Only quadrant where cosine is nagative,and tangent is positive is quadrant III.
sin(theta)= + OR - sqroot[1-cos^2(theta)]
In quadrant III sine is negative so:
sin(theta)= - sqroot[1-cos^2(theta)]
sin(theta)= - sqroot[1-(-1/2)^2]
sin(theta)= - sqroot(1-1/4)
sin(theta)= - sqroot(3/4)
sin(theta)= - sqroot(3) /2
tan(theta)=sin(theta)/cos(theta)=
[-sqroot(3)/2]/(-1/2)= + sqroot(3)=
sqroot(3)
ctg(theta)=1/tan(theta)=1/sqroot(3)
sec(theta)=1/cos(theta)=1/(-1/2)= -2
cosec(theta)=1/sin(theta)=1/[-sqroot(3) /2]= -2/sqroot(3)
determine all trig functions of theta using the given info, state the answers correct to the nearest hundredth.
cos (theta) = -5/13; tan (theta) < 0
To determine the quadrant that contains the terminal side of 0, we look at the given information:
- The cosine of 0 is -1/2, which means that the x-coordinate of the point on the unit circle corresponding to 0 is -1/2.
- The tangent of 0 is positive, which means that the y-coordinate of the point on the unit circle corresponding to 0 is positive.
From these conditions, we can infer that the point is located in the 2nd quadrant.
Now, let's find the exact values of the other five trig functions of 0:
1. Sine (sin): Since the sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle, we can find it using the Pythagorean identity:
sin^2(0) + cos^2(0) = 1
sin^2(0) + (-1/2)^2 = 1
sin^2(0) + 1/4 = 1
sin^2(0) = 1 - 1/4 = 3/4
Taking the square root of both sides:
sin(0) = √(3/4)
sin(0) = √3/2
2. Cosecant (csc): The cosecant of an angle is the reciprocal of the sine:
csc(0) = 1/sin(0)
csc(0) = 1 / (√3/2)
csc(0) = 2/√3 * √3/√3
csc(0) = 2√3 / 3
3. Secant (sec): The secant of an angle is the reciprocal of the cosine:
sec(0) = 1/cos(0)
sec(0) = 1 / (-1/2)
sec(0) = -2
4. Cosecant (cosec): The cotangent of an angle is the reciprocal of the tangent:
cot(0) = 1/tan(0)
cot(0) = 1 / tan(0)
cot(0) = 1 / (√3/2)
cot(0) = 2/√3 * √3/√3
cot(0) = 2√3 / 3
5. Cotangent (cot): The cotangent of an angle is the reciprocal of the tangent:
cot(0) = 1/tan(0)
cot(0) = 1 / (√3/2)
cot(0) = 2/√3 * √3/√3
cot(0) = 2√3 / 3
So the exact values of the other five trig functions of 0 are:
- sin(0) = √3/2
- csc(0) = 2√3 / 3
- sec(0) = -2
- cosec(0) = 2√3 / 3
- cot(0) = 2√3 / 3
To find the quadrant that contains the terminal side of θ, we can use the information that cos θ = -1/2 and tan θ > 0.
First, let's find the angle θ for cos θ = -1/2. We know that cos θ = adjacent/hypotenuse, so in a right triangle, the adjacent side will be -1 (since cos is negative in the second and third quadrants) and the hypotenuse will be 2 (since cos θ = -1/2). Using the Pythagorean theorem, we can find the length of the opposite side:
opposite^2 = hypotenuse^2 - adjacent^2
opposite^2 = 2^2 - (-1)^2
opposite^2 = 4 - 1
opposite^2 = 3
opposite = √3
So, in the second quadrant, the angle θ will be such that cos θ = -1/2 and √3 is the length of the opposite side. To find the exact value of sin θ, we know that sin θ = opposite/hypotenuse, so sin θ = √3/2.
Now, let's find the other five trigonometric function values for the angle θ:
- The secant function is the reciprocal of the cosine function, so sec θ = 1/cos θ = 1/(-1/2) = -2.
- The cosecant function is the reciprocal of the sine function, so csc θ = 1/sin θ = 1/(√3/2) = 2/√3 = (2√3)/3.
- The tangent function is the sine divided by the cosine, so tan θ = sin θ / cos θ = (√3/2) / (-1/2) = -√3.
- The cotangent function is the reciprocal of the tangent function, so cot θ = 1/tan θ = 1/(-√3) = -1/√3 = -√3/3.
- The cosecant function is the reciprocal of the cosine function, so sec θ = 1/cos θ = 1/(-1/2) = -2.
Therefore, the values of the other five trigonometric functions for the angle θ are:
sin θ = √3/2
cos θ = -1/2
sec θ = -2
csc θ = (2√3)/3
tan θ = -√3
cot θ = -√3/3
We have found the exact values of the other five trigonometric functions for the given angle θ.