If sin x=-3/5 and sec x>0, find all the other circular function values of x
sin x=-3/5 and sec x>0 ---> sin x=-3/5 and cos x>0
so your angle must be in quadrants IV
If you sketch your triangle , you will realize that you are dealing with the 3-4-5 right-angled triangle, and using the CAST rule
sinx = -3/5 , cscx = -5/3
cosx = 4/5, secx = 5/4
tanx = -3/4, cotx = -4/3
To find the other circular function values of x, we need to determine the values of cosine, tangent, cosecant, cotangent, and secant.
Given that sin(x) = -3/5 and sec(x) > 0, we can solve for cosine using the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Step 1: Find the value of cosine (cos(x)):
Since sin(x) = -3/5, we can use the Pythagorean identity to solve for cos(x):
(-3/5)^2 + cos^2(x) = 1
9/25 + cos^2(x) = 1
cos^2(x) = 1 - 9/25
cos^2(x) = 25/25 - 9/25
cos^2(x) = 16/25
Taking the square root of both sides, we have:
cos(x) = ±√(16/25)
cos(x) = ±(4/5)
Note: We take the positive value of cosine because sec(x) is positive.
Step 2: Find the value of tangent (tan(x)):
We can calculate tangent using the formula: tan(x) = sin(x) / cos(x)
tan(x) = (-3/5) / (4/5)
tan(x) = -3/4
Step 3: Find the value of cosecant (csc(x)):
We know that cosecant is the reciprocal of sine, so csc(x) = 1/sin(x)
csc(x) = 1 / (-3/5)
csc(x) = -5/3
Step 4: Find the value of cotangent (cot(x)):
We can calculate cotangent using the formula: cot(x) = 1/tan(x)
cot(x) = 1 / (-3/4)
cot(x) = -4/3
Step 5: Find the value of secant (sec(x)):
Given that sec(x) > 0, we already know that sec(x) = 4/5, since secant is the reciprocal of cosine.
To summarize, the circular function values for x are:
sin(x) = -3/5
cos(x) = 4/5
tan(x) = -3/4
csc(x) = -5/3
cot(x) = -4/3
sec(x) = 4/5