Let be an angle in quadrant II such that sec(theta) -4/3 .
Find the exact values of cot (theta) and sin (theta).
cos(theta)=1/sec(theta)= -3/4
sin(theta)+ OR - sqroot[1-cos^2(theta)]
Sine is positive in Quadrant II so:
sin(theta)=sqroot[1-(-3/4)^2]
sin(theta)=sqroot(1-9/16)
sin(theta)=sqroot(16/16-9/16)
sin(theta)=sqroot(7/16)
sin(theta = sqroot(7) / 4
cot(theta)=cos(theta)/sin(theta)
cot(theta)=(-3/4)/(sqroot(7)/4)
cot(theta)= -3/sqroot(7)
To find the exact values of cot(theta) and sin(theta) given that sec(theta) = -4/3, we can use the following trigonometric relationship:
sec^2(theta) = 1 + tan^2(theta)
Since we know that sec(theta) = -4/3, we can substitute this value into the equation:
(-4/3)^2 = 1 + tan^2(theta)
Simplifying, we have:
16/9 = 1 + tan^2(theta)
Next, we can solve for tan^2(theta) by subtracting 1 from both sides:
tan^2(theta) = 16/9 - 1
tan^2(theta) = 16/9 - 9/9
tan^2(theta) = 7/9
Now, taking the square root of both sides, we get:
tan(theta) = ±√(7/9)
Since we are given that theta is in quadrant II (where cosine is negative), tangent is positive in the second quadrant. Therefore:
tan(theta) = √(7/9)
Now, we can use the Pythagorean identity to find the value of sin(theta):
sin^2(theta) = 1 - cos^2(theta)
Since cos(theta) is equal to -3/4 (since sec(theta) = -4/3), we can substitute this value into the equation:
sin^2(theta) = 1 - (-3/4)^2
sin^2(theta) = 1 - 9/16
sin^2(theta) = 16/16 - 9/16
sin^2(theta) = 7/16
Again, taking the square root of both sides, we have:
sin(theta) = ±√(7/16)
Since theta is in quadrant II where sine is positive, we have:
sin(theta) = √(7/16)
Therefore, the exact values of cot(theta) and sin(theta) are:
cot(theta) = 1/tan(theta) = 1/√(7/9) = √(9/7)
sin(theta) = √(7/16)
To find the exact values of cot(theta) and sin(theta) given that sec(theta) = -4/3 and theta is in quadrant II, we can use the definitions of the trigonometric functions and the relationship between them.
Since sec(theta) = -4/3, we know that cos(theta) = -3/4. In quadrant II, cosine is negative.
Now, we can use the Pythagorean identity to find sin(theta):
sin^2(theta) + cos^2(theta) = 1
sin^2(theta) + (-3/4)^2 = 1
sin^2(theta) + 9/16 = 1
sin^2(theta) = 1 - 9/16
sin^2(theta) = 16/16 - 9/16
sin^2(theta) = 7/16
Taking the square root of both sides:
sin(theta) = ±√(7/16)
Since theta is in quadrant II, sin(theta) is positive. Therefore,
sin(theta) = √(7/16) = √7/4
Next, we can find cot(theta) using the definitions of cotangent and tangent:
cot(theta) = 1/tan(theta)
Since sec(theta) = -4/3, we know that tan(theta) = 1/(-4/3) = -3/4.
cot(theta) = 1/(-3/4) = -4/3
So, the exact values of cot(theta) and sin(theta) are cot(theta) = -4/3 and sin(theta) = √7/4.