1) evaluate without a calculator: a)sin(3.14/4) b) cos(-3(3.14)/4) c) tan(4(3.14)/3) d) arccos(- square root of three/2) e) csctheata=2
2) verify the following identities: a) cotxcosx+sinx=cscx b)[(1+sinx)/ cosx] + [cosx/ (1+sinx)]= 2secx c) (sin^2x + 2sinx+1)/cos^2x= (1+sinx)/(1-sinx)
You should memorize the following diagrams:
1. the 30°-60°-90° triangle with matching sides
1 -- √3 -- 2 (angles in radians, π/6 π/3 π/2)
2. the 45° -- 45° -- 90° triangle with corresponding sides 1 -- 1 -- √2
So sin π/4 or sin 45° = 1/√2
arccos(-√3/2) = .....
I know from looking at my triangle that cos60° or cosπ/3 = √3/2
Also I know that the cosine is negative in II and III
so arccos(-√3/2) = 180-60 = 120° or 2π/3
cscØ = 2
sinØ = 1/2
Ø = 30° or π/6 , looking at my triangle
With identities, it is usually a good idea to change all trig ratios to sines and cosines, and let the chips fall as they may.
I will do the first one:
LS = cotx cosx + sinx
= (cosx/sinx)(cosx) + sinx
= (cos^2 x + sin^2 x)/sinx
= 1/sinx
= csc x
= RS
try the others using that concept, let me know if you were successful.
1) To evaluate the trigonometric expressions without a calculator, you need to use special angles and trigonometric identities.
a) sin(3.14/4):
We know that sin(45 degrees) = sin(pi/4) = 1/sqrt(2). So, sin(3.14/4) = 1/sqrt(2).
b) cos(-3(3.14)/4):
We can use the cofunction identity to simplify this expression. The cofunction of cos(x) is sin(90 degrees - x). In this case, cos(-3(3.14)/4) = sin(90 degrees + 3(3.14)/4) = sin(135 degrees).
By using the special angle 45 degrees, we can rewrite sin(135 degrees) as sin(135 degrees - 90 degrees). Then, we have sin(45 degrees) = 1/sqrt(2).
c) tan(4(3.14)/3):
To evaluate tan(4(3.14)/3), we convert the tangent function into the ratio of sin and cos. tan(x) = sin(x)/cos(x).
Using the cofunction identity, tan(60 degrees) = 1/sqrt(3). So, tan(4(3.14)/3) = sin(4(3.14)/3)/cos(4(3.14)/3) = sin(240 degrees)/cos(240 degrees) = 1/sqrt(3).
d) arccos(-sqrt(3)/2):
We can use the inverse cosine function identity to evaluate this expression. The inverse cosine function of -sqrt(3)/2 is equal to 5pi/6 or 150 degrees.
e) csc(theta) = 2:
To find the value of theta when csc(theta) = 2, we can use the reciprocal identity csc(theta) = 1/sin(theta). So, sin(theta) = 1/2.
Using the special angle 30 degrees, we have sin(30 degrees) = 1/2. Therefore, theta can be 30 degrees or pi/6 radians.
2) To verify the given trigonometric identities, we need to simplify both sides of the equation and show that they are equal.
a) cot(x)cos(x) + sin(x) = csc(x):
We start by simplifying the left-hand side (LHS):
cot(x) = cos(x)/sin(x), so LHS = cos(x)/sin(x) * cos(x) + sin(x) = cos^2(x)/sin(x) + sin(x).
To simplify the right-hand side (RHS):
csc(x) = 1/sin(x), so RHS = 1/sin(x).
To make the denominators the same, we multiply the RHS by sin(x)/sin(x):
RHS = sin(x)/sin^2(x).
Now, we can rewrite the LHS:
LHS = cos^2(x)/sin(x) + sin(x) = (1 - sin^2(x))/sin(x) + sin(x) = 1/sin(x) - sin^2(x)/sin(x) + sin(x) = 1/sin(x).
Therefore, the LHS = RHS, and the identity cot(x)cos(x) + sin(x) = csc(x) is verified.
b) [(1 + sin(x))/cos(x)] + [cos(x)/(1 + sin(x))] = 2sec(x):
First, simplify each term: [(1 + sin(x))/cos(x)] = sec(x) and [cos(x)/(1 + sin(x))] = sec(x).
Now we have sec(x) + sec(x) = 2sec(x), which satisfies the given identity.
c) (sin^2(x) + 2sin(x) + 1)/cos^2(x) = (1 + sin(x))/(1 - sin(x)):
Starting with the left-hand side (LHS):
(sin^2(x) + 2sin(x) + 1)/cos^2(x) = (sin(x) + 1)^2/(1 - sin^2(x)).
Using the Pythagorean Identity (1 - sin^2(x) = cos^2(x)), we can simplify it to:
= (sin(x) + 1)^2/cos^2(x).
Now, simplify the right-hand side (RHS):
(1 + sin(x))/(1 - sin(x)) = [(1 + sin(x))^2]/[(1 - sin(x))(1 + sin(x))] = (sin(x) + 1)^2/(1 - sin^2(x)).
By simplifying both sides, we can see that the LHS = RHS, and the identity is verified.
1) Evaluate without a calculator:
a) sin(3.14/4)
To evaluate sin(3.14/4), we need to use the value of sin(pi/4) which is equal to 1/sqrt(2) or approximately 0.707.
b) cos(-3(3.14)/4)
To evaluate cos(-3(3.14)/4), we need to use the value of cos(-3pi/4) which is equal to -1/sqrt(2) or approximately -0.707.
c) tan(4(3.14)/3)
To evaluate tan(4(3.14)/3), we need to use the value of tan(4pi/3), which is equal to -sqrt(3). Hence, tan(4(3.14)/3) = -sqrt(3).
d) arccos(- sqrt(3)/2)
To evaluate arccos(-sqrt(3)/2), we need to find the angle whose cosine is -sqrt(3)/2. This angle is 5pi/6 radians or 150 degrees.
e) csc(theta) = 2
To solve csc(theta) = 2, we need to find the angle whose cosecant is 2. This angle is approximately 30 degrees or pi/6 radians.
2) Verify the following identities:
a) cot(x)cos(x) + sin(x) = csc(x)
Starting with the left side:
cot(x)cos(x) + sin(x)
= (cos(x)/sin(x)) * cos(x) + sin(x)
= (cos^2(x))/sin(x) + sin(x)
= (1 - sin^2(x))/sin(x) + sin(x) [Using identity cos^2(x) = 1 - sin^2(x)]
= 1/sin(x) - (sin^2(x))/sin(x) + sin(x)
= 1/sin(x) - sin(x) + sin(x)
= 1/sin(x)
= csc(x)
Hence, the identity cot(x)cos(x) + sin(x) = csc(x) is verified.
b) [(1 + sin(x))/cos(x)] + [cos(x)/(1 + sin(x))] = 2sec(x)
Starting with the left side:
[(1 + sin(x))/cos(x)] + [cos(x)/(1 + sin(x))]
= [(1 + sin(x))*(1 + sin(x))) + (cos^2(x))]/[cos(x)*(1 + sin(x))] [Using a common denominator]
= (1 + 2sin(x) + sin^2(x) + cos^2(x))/(cos(x) + cos(x)sin(x))
= (2 + 2sin(x))/(cos(x) + cos(x)sin(x))
= (2(1 + sin(x)))/(cos(x)(1 + sin(x)))
= 2/(cos(x))
= 2sec(x)
Hence, the identity [(1+sin(x))/cos(x)] + [cos(x)/(1+sin(x))] = 2sec(x) is verified.
c) (sin^2(x) + 2sin(x) + 1)/cos^2(x) = (1 + sin(x))/(1 - sin(x))
Starting with the left side:
(sin^2(x) + 2sin(x) + 1)/cos^2(x)
= (sin^2(x) + 2sin(x) + 1)/(1 - sin^2(x)) [Using identity cos^2(x) = 1 - sin^2(x)]
= (sin^2(x) + 2sin(x) + 1)/[(1 - sin(x))(1 + sin(x))]
= (sin^2(x) + 2sin(x) + 1)/(1 - sin^2(x))
= (sin^2(x) + 2sin(x) + 1)/(1 - sin^2(x))
= (sin^2(x) + 2sin(x) + 1)/(1 - sin^2(x))
= (1 + sin(x))/(1 - sin(x))
Hence, the identity (sin^2(x) + 2sin(x) + 1)/cos^2(x) = (1 + sin(x))/(1 - sin(x)) is verified.