Using the exact trigonometric ratios of 45 degree,30 degree and 60 degree,evaluate the following,leaving where necessary your answer in surd form. (a) 3 sin 30 degree-2 cos 60 degree (b)(tan 45 degree)(4 cos 60 degree-2 cos 60 degree)
sin 30 = 1/2
thus, we get 3/2 - 2cos 60
cos 60 = 1/2
thus we get 3/2 - 2/2 = 1/2
tan45 = 1
so the equation is just 4cos60 - 2cos60
cos 60 = 1/2, as previously mentioned
thus, we get 2 - 1 = 1.....
btw...I think you can just plug these functions into your calculator to figure these out, it is not too hard.
To evaluate the expressions using the exact trigonometric ratios of 45 degrees, 30 degrees, and 60 degrees, we need to know the values of sin(30°), cos(60°), sin(45°), and cos(45°). Here are the values:
sin(30°) = 1/2
cos(60°) = 1/2
sin(45°) = √2/2
cos(45°) = √2/2
Now let's solve the given expressions:
(a) 3 sin 30° - 2 cos 60°:
Replace sin(30°) and cos(60°) with their respective values:
3 * (1/2) - 2 * (1/2)
Multiply the numerators and denominators together:
3/2 - 2/2
Combine the fractions:
(3 - 2)/2
Simplify the numerator:
1/2
So the answer to (a) is 1/2.
(b) (tan 45°)(4 cos 60° - 2 cos 60°):
Replace tan(45°), cos(60°), and cos(60°) with their respective values:
(√2/2) * (4 * (1/2) - 2 * (1/2))
Simplify inside the parentheses:
(√2/2) * (2 - 1)
Multiply the numerators and denominators together:
(2√2/2) * (1/1)
Cancel out the common factor of 2 in the numerator and denominator:
(√2/1) * (1/1)
Simplify:
√2
So the answer to (b) is √2.