Given cos x = -3/5 , sin y =12/13 and 0º< x <360º,x and y are on the same quadrant. Without using mathematical table and scientific calculator,find the values for:
(a)sin 1/2 x
(b)cos (x+y)
since cosx < 0 and siny > 0, and both x and y are in the same quadrants, the angles must be in II
if cosx = -3/5, then sinx = 4/5
if siny = 12/13, then cosy = - 5/13
a) from cos 2A = 1 - 2 cos^2 A
cos x = 1 - 2 sin^2 (x/2)
-3/5 = 1 - 2sin^2 (x/2)
2sin^2 (x/2) = 1 + 3/5 = 8/5
sin^2 (x/2) = 4/5
sin (x/2) = 2/√5
b)
cos(x+y) = cosxcosy - sinxsiny
= sub in the above values and simplify
To find the values of the trigonometric functions without using a mathematical table or scientific calculator, we can use the following identities:
1. sin(2θ) = 2sin(θ)cos(θ)
2. cos(2θ) = cos²(θ) - sin²(θ)
Using the given information, cos(x) = -3/5 and sin(y) = 12/13, we can determine the values of x and y.
(a) sin(1/2x):
To find sin(1/2x), we need to first find the value of x/2. Since we know the value of cos(x), we can use the inverse cosine function to find x:
cos(x) = -3/5
x = cos^(-1)(-3/5)
Based on the given information, we know that x is in the same quadrant as y. Since cos is negative in the given quadrant, x will be in the 2nd or 3rd quadrant. Knowing this quadrant restriction will help us to determine the proper value for x.
(b) cos(x+y):
To find cos(x+y), we need to know the values of both x and y. From the given information, we only know the value of cos(x) and sin(y). However, we can use the identity cos²(θ) + sin²(θ) = 1 to find sin(x) and cos(y).
cos²(θ) + sin²(θ) = 1
cos²(x) + sin²(x) = 1
sin²(x) = 1 - cos²(x)
sin(x) = ±√(1 - cos²(x))
With these values in hand, we can now calculate cos(x+y) using the identity:
cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
Now, let's go through the steps to find the values of sin(1/2x) and cos(x+y) without using a mathematical table or scientific calculator.