Use the value sin x = 1/ 2 and cos x > 0 to find the values of all six trigonometric functions.
Leave your answer in exact form, not in decimal
construct your right-angled triangle.
r^2 = x^2 + y^2 , (x is a side , not an angle)
2^2 = x^2 + 1^2 , since sinØ = opposite/hypoenuse or y/r
x^2 = 3
x = √3 , we know the angle must be in Quad I
sinx = 1/2 --- given
cscx = 2
cosx = √3/2
secx = 2/√3
tanx = 1/√3
cotx = √3
by the way, you angle x would be π/6 or 30°
To find the values of all six trigonometric functions using the given information, we will need to refer to the unit circle.
Let's start by representing the given information on the unit circle:
Since sin(x) = 1/2, we know that the angle x corresponds to a point on the unit circle where the y-coordinate is 1/2.
Since cos(x) > 0, we know that the angle x corresponds to a point on the right side of the unit circle, where the x-coordinate is positive.
Now let's find the values of the trigonometric functions:
1. Sine (sin(x)): We already know that sin(x) = 1/2, so the value of sin(x) is 1/2.
2. Cosine (cos(x)): To find the value of cos(x), we can use the Pythagorean identity sin²(x) + cos²(x) = 1. Since we know sin(x) = 1/2, we can substitute this value into the identity:
(1/2)² + cos²(x) = 1
1/4 + cos²(x) = 1
cos²(x) = 3/4
Since cos(x) > 0, we take the positive square root:
cos(x) = √(3/4) = √3/2
3. Tangent (tan(x)): The tangent is calculated by dividing the sine by the cosine:
tan(x) = sin(x) / cos(x) = (1/2) / (√3/2) = 1/√3 = √3/3
4. Cosecant (csc(x)): The cosecant is the reciprocal of the sine:
csc(x) = 1 / sin(x) = 1 / (1/2) = 2/1 = 2
5. Secant (sec(x)): The secant is the reciprocal of the cosine:
sec(x) = 1 / cos(x) = 1 / (√3/2) = 2 / √3 = (2√3) / 3
6. Cotangent (cot(x)): The cotangent is the reciprocal of the tangent:
cot(x) = 1 / tan(x) = 1 / (√3/3) = √3/3
So, the values of all six trigonometric functions are:
sin(x) = 1/2
cos(x) = √3/2
tan(x) = √3/3
csc(x) = 2
sec(x) = (2√3) / 3
cot(x) = √3/3