find the 5 other thrig functions if cos(theta) = square root of 2/2 and cotangent is less than 0
Given:CosA=SQRT2/2
Since the cosine is positive and the
tangent and cotangent are negative
(less than 0), we are in the 4th Quad-
rant. CosA=sqrt2/2=x/r, x=sqrt2, r=2,
Y=sqrt(r^2-x^2)=sqrt(4-2)=+/-sqrt2,
Y=-sqrt2(4th Quadrant).
SinA=y/r=-sqrt2/2.
CosA=sqrt2/2=x/r(Given)
TanA=y/x=-sqrt2/sqrt2=-1
cscA=-(2/sqrt2).
secA=2/sqrt2
CotanA=sqrt2/-sqrt2=-1
To find the 5 other trigonometric functions, we first need to determine the value of sine (sin), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
Given that cos(theta) = √2/2 and cot(theta) < 0, we can use the Pythagorean identity to find the value of sin(theta).
Step 1: Find sin(theta)
Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can substitute the known value of cos(theta) into the equation:
sin^2(theta) + (√2/2)^2 = 1
sin^2(theta) + 2/4 = 1
sin^2(theta) + 1/2 = 1
sin^2(theta) = 1 - 1/2
sin^2(theta) = 1/2
sin(theta) = ±√(1/2)
Since sine is positive in the first and second quadrants, sin(theta) = √(1/2) = √2/2.
Step 2: Find tan(theta)
tan(theta) = sin(theta) / cos(theta)
tan(theta) = (√2/2) / (√2/2)
tan(theta) = 1
Step 3: Find csc(theta)
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (√2/2)
csc(theta) = 2 / √2
To rationalize the denominator, we multiply both the numerator and denominator by √2:
csc(theta) = (2√2) / (2)
csc(theta) = √2
Step 4: Find sec(theta)
sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (√2/2)
sec(theta) = 2 / √2
To rationalize the denominator, we multiply both the numerator and denominator by √2:
sec(theta) = (2√2) / (2)
sec(theta) = √2
Step 5: Find cot(theta)
cot(theta) = 1 / tan(theta)
cot(theta) = 1 / 1
cot(theta) = 1
Therefore, the values of the 5 other trigonometric functions, given cos(theta) = √2/2 and cot(theta) < 0, are:
sin(theta) = √2/2
tan(theta) = 1
csc(theta) = √2
sec(theta) = √2
cot(theta) = 1