a. Find the exact value of
sin (theta - pi/6)
b. Find the exact value of
cos (theta - pi/4)
c. For what numbers x, -2pi less than or equal to x less than or equal to 2pi does cos x = -1
d. What is the range of the cosine function?
e. For what numbers x, -pi less than or equal to x less than or equal to pi is the graph of y = cos x increasing?
f. Given: tan alpha = -4/3,
pi/2 < alpha , pi and
cos beta = 1/2, 3pi/2 < beta , 2 pi.
Find: tan (alpha-beta)
a.
sin(-theta)= -sin(theta)
sin(-pi/6)= - sin(pi/6)= -1/2
b.
cos(-theta)=cos(theta)
cos(-pi/6)=cos(pi/6)= sqroot(3)/2
c.
-pi
d.
The cosine function, like all of the trigonometric functions, is periodic about the rotation around a circle. Since the cosine is defined as the adjacent/hypotenuse of a right triangle, you can clearly see that its value can never be greater than one or less than -1 since the hypotenuse is always longer than the adjacent side. It turns out that, indeed, the cosine's range is from -1 to 1, written [-1,1].
e.
-pi/2,pi
a. To find the exact value of sin(theta - pi/6), you can use the trigonometric identity for the difference of angles:
sin(theta - phi) = sin(theta) * cos(phi) - cos(theta) * sin(phi).
In this case, phi = pi/6.
sin(theta - pi/6) = sin(theta) * cos(pi/6) - cos(theta) * sin(pi/6).
b. To find the exact value of cos(theta - pi/4), you can use the same trigonometric identity as above:
cos(theta - phi) = cos(theta) * cos(phi) + sin(theta) * sin(phi).
In this case, phi = pi/4.
cos(theta - pi/4) = cos(theta) * cos(pi/4) + sin(theta) * sin(pi/4).
c. To find the numbers x in the range -2pi ≤ x ≤ 2pi where cos(x) = -1, you need to consider that cosine is equal to -1 at the angles π and 2π. Therefore, the values of x satisfying cos(x) = -1 are x = π and x = 2π.
d. The range of the cosine function is [-1, 1]. This means that the output values of cosine can range from -1 (the minimum value) to 1 (the maximum value).
e. The graph of y = cos(x) is increasing in the interval -π/2 ≤ x ≤ π/2. In this interval, the cosine function starts at its minimum value of -1 and increases up to its maximum value of 1.
f. To find tan(alpha - beta), you can use the trigonometric identity for the difference of angles:
tan(alpha - beta) = (tan(alpha) - tan(beta)) / (1 + tan(alpha) * tan(beta)).
Given that tan(alpha) = -4/3 and cos(beta) = 1/2, you can use the Pythagorean identity to find tan(beta):
tan(beta) = sin(beta) / cos(beta) = sqrt(1 - cos^2(beta)) / cos(beta) = sqrt(1 - (1/2)^2) / (1/2) = √3 / 2.
Now, substitute the values into the formula for tan(alpha - beta):
tan(alpha - beta) = (-4/3 - √3/2) / (1 + (-4/3) * (√3/2)).
Simplify and calculate the value.
a. To find the exact value of sin(theta - pi/6), you can use the trigonometric identity for the difference of angles:
sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)
In this case, A = theta and B = pi/6. Since we know the values of sine and cosine for pi/6, we can substitute them into the formula:
sin(theta - pi/6) = sin(theta) * cos(pi/6) - cos(theta) * sin(pi/6)
b. Similarly, to find the exact value of cos(theta - pi/4), you can use the same trigonometric identity:
cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
In this case, A = theta and B = pi/4. Substitute the known values:
cos(theta - pi/4) = cos(theta) * cos(pi/4) + sin(theta) * sin(pi/4)
c. To find the values of x for which cos(x) = -1, we need to recall the unit circle and the values of cosine on it. Cosine is equal to -1 at the angles π (pi) and 2π (2pi), which means the solutions lie in the interval -2π ≤ x ≤ 2π.
d. The range of the cosine function is the set of all possible values that the function can take. Since cosine is a periodic function with a period of 2π, its range is bounded between -1 and 1. In other words, -1 ≤ cos(x) ≤ 1 for all x.
e. To determine when the graph of y = cos(x) is increasing, we need to look at the derivative of the function. The derivative of cos(x) is -sin(x). When -pi ≤ x ≤ pi, sin(x) is positive, so the derivative is negative. This means that the function is decreasing in this interval. However, when -pi < x < pi, the graph of y = cos(x) is increasing.
f. To find the value of tan(alpha - beta), we first need to recall the trigonometric identity for the difference of angles:
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) * tan(B))
In this case, A = alpha and B = beta. Given that tan(alpha) = -4/3 and tan(beta) = 1/2, we have:
tan(alpha - beta) = (tan(alpha) - tan(beta)) / (1 + tan(alpha) * tan(beta))
Substituting the known values:
tan(alpha - beta) = (-4/3 - 1/2) / (1 + (-4/3) * (1/2))