I need help with these kind of problems!
Can someone tell me how to work this problem out?
Given tan0=2 square root of 5 over 4, and sin0<0.
a. cos0 b. csc0
The 0 after tan, sin, cos, and csc stands for angle. The 0 after the < stands for negative or less than. I know it sounds confusing, but if somebody knows a great web site to get help from, please inform me!
To find the values of cos0 and csc0 given the information provided, we can use the trigonometric identity:
sin^2θ + cos^2θ = 1
Given that sin0 < 0, we know that the angle is in the third or fourth quadrant, where cosine is negative.
Step 1: Find sin0
We're given that tan0 = 2√5/4. We can use the fact that tanθ = sinθ/cosθ:
tan0 = sin0/cos0
sin0 = tan0 * cos0
We know that tan0 = 2√5/4, so:
sin0 = (2√5/4) * cos0
Step 2: Solve for cos0
Using the Pythagorean identity mentioned earlier:
sin^2θ + cos^2θ = 1
(sin0)^2 + (cos0)^2 = 1
Substituting the value of sin0 obtained in Step 1:
[(2√5/4) * cos0]^2 + (cos0)^2 = 1
(4√5/4 * cos0)^2 + (cos0)^2 = 1
Simplifying:
(20/16) * (cos0)^2 + (cos0)^2 = 1
(20/16 + 1) * (cos0)^2 = 1
20/16 + 1 = 36/16
(36/16) * (cos0)^2 = 1
(9/4) * (cos0)^2 = 1
(cos0)^2 = 4/9
Taking the square root of both sides:
cos0 = ± √(4/9)
Since cos0 is negative in the third and fourth quadrants:
cos0 = -√(4/9) = -2/3
So the answer to part a is cos0 = -2/3.
Step 3: Solve for csc0
Using the identity cscθ = 1/sinθ:
csc0 = 1/sin0
We obtained the value of sin0 in Step 1:
csc0 = 1/[(2√5/4) * cos0]
Simplifying:
csc0 = 4 / [2√5 * cos0]
csc0 = 2 / (√5 * cos0)
Substituting the value of cos0 obtained in Step 2:
csc0 = 2 / (√5 * (-2/3))
Simplifying:
csc0 = -6 / (2√5)
csc0 = -3 / √5
To rationalize the denominator, we can multiply both the numerator and denominator by √5:
csc0 = (-3/√5) * (√5/√5)
csc0 = -3√5 / 5
So the answer to part b is csc0 = -3√5 / 5.