Evaluate each expression without using a calculator.
77. cos Tan^-1(-2))
79. sin (2tan^-1 12/5)
83. sin(2 cos^-1 1/5)
87. cos (tan ^-1 5/12 -tan^-1 3/4)
89. sin (sin-1 1/2 + tan^-1(-3))
To evaluate these expressions without using a calculator, we will use trigonometric identities and basic trigonometric relationships.
77. cos(Tan^(-1)(-2)):
Let's first find the value of Tan^(-1)(-2). This means finding an angle whose tangent is -2. We know that Tan(x) = opposite/adjacent, so we have:
Tan(x) = -2
Taking the inverse tangent of both sides, we get:
x = Tan^(-1)(-2)
To find the value of cos(Tan^(-1)(-2)), we can use the identity:
cos(Tan^(-1)(x)) = 1 / sqrt(1 + x^2)
In this case, x = -2, so we have:
cos(Tan^(-1)(-2)) = 1 / sqrt(1 + (-2)^2)
= 1 / sqrt(1 + 4)
= 1 / sqrt(5)
= sqrt(5) / 5
Therefore, cos(Tan^(-1)(-2)) = sqrt(5) / 5.
79. sin(2Tan^(-1)(12/5)):
Again, we need to find the value of Tan^(-1)(12/5). This means finding an angle whose tangent is 12/5. We have:
Tan(x) = 12/5
Taking the inverse tangent of both sides, we get:
x = Tan^(-1)(12/5)
To evaluate sin(2Tan^(-1)(12/5)), we can use the identity:
sin(2x) = 2sin(x)cos(x)
In this case, x = Tan^(-1)(12/5), so we have:
sin(2Tan^(-1)(12/5)) = 2sin(Tan^(-1)(12/5))cos(Tan^(-1)(12/5))
Using the definitions of sine and cosine, we can write:
sin(2Tan^(-1)(12/5)) = 2(12/13)(5/13)
= 120/169
Therefore, sin(2Tan^(-1)(12/5)) = 120/169.
83. sin(2cos^(-1)(1/5)):
To evaluate this expression, let's first find the value of cos^(-1)(1/5). This means finding an angle whose cosine is 1/5. We have:
cos(x) = 1/5
Taking the inverse cosine of both sides, we get:
x = cos^(-1)(1/5)
To simplify sin(2cos^(-1)(1/5)), we can use the identity:
sin(2x) = 2sin(x)cos(x)
In this case, x = cos^(-1)(1/5), so we have:
sin(2cos^(-1)(1/5)) = 2sin(cos^(-1)(1/5))cos(cos^(-1)(1/5))
Using the definitions of sine and cosine, we can write:
sin(2cos^(-1)(1/5)) = 2(4/5)(1/5)
= 8/25
Therefore, sin(2cos^(-1)(1/5)) = 8/25.
87. cos(Tan^(-1)(5/12) - Tan^(-1)(3/4)):
Let's find the value of Tan^(-1)(5/12) and Tan^(-1)(3/4).
Tan^(-1)(5/12) means finding an angle whose tangent is 5/12. We have:
Tan(x) = 5/12
Taking the inverse tangent of both sides, we get:
x = Tan^(-1)(5/12)
Similarly, for Tan^(-1)(3/4), we have:
Tan(x) = 3/4
x = Tan^(-1)(3/4)
Now, to evaluate cos(Tan^(-1)(5/12) - Tan^(-1)(3/4)), we can use the identity:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
In this case, a = Tan^(-1)(5/12) and b = Tan^(-1)(3/4), so we have:
cos(Tan^(-1)(5/12) - Tan^(-1)(3/4)) = cos(Tan^(-1)(5/12))cos(Tan^(-1)(3/4)) + sin(Tan^(-1)(5/12))sin(Tan^(-1)(3/4))
Using the definitions of sine and cosine, we get:
cos(Tan^(-1)(5/12) - Tan^(-1)(3/4)) = (12/13)(4/5) + (5/13)(3/4)
= 48/65 + 15/52
= (48·52 + 15·65) / (65·52)
= 2496/3380
= 624/845
Therefore, cos(Tan^(-1)(5/12) - Tan^(-1)(3/4)) = 624/845.
89. sin(sin^(-1)(1/2) + Tan^(-1)(-3)):
Let's find the value of sin^(-1)(1/2) and Tan^(-1)(-3).
sin^(-1)(1/2) means finding an angle whose sine is 1/2. We have:
sin(x) = 1/2
Taking the inverse sine of both sides, we get:
x = sin^(-1)(1/2)
For Tan^(-1)(-3), it means finding an angle whose tangent is -3. We have:
Tan(x) = -3
Taking the inverse tangent of both sides, we get:
x = Tan^(-1)(-3)
Now, to evaluate sin(sin^(-1)(1/2) + Tan^(-1)(-3)), we can use the identity:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case, a = sin^(-1)(1/2) and b = Tan^(-1)(-3), so we have:
sin(sin^(-1)(1/2) + Tan^(-1)(-3)) = sin(sin^(-1)(1/2))cos(Tan^(-1)(-3)) + cos(sin^(-1)(1/2))sin(Tan^(-1)(-3))
Using the definitions of sine and cosine, we get:
sin(sin^(-1)(1/2) + Tan^(-1)(-3)) = (1/2)(-3/√(3^2 + 1^2)) + (√(3^2 + 1^2))(1/2)
= -3/2√10 + 4/2√10
= 1/2√10
Therefore, sin(sin^(-1)(1/2) + Tan^(-1)(-3)) = 1/2√10.