Question

find sin x/2 , cos x/2 , and tan x/2 from the given information. sec(x) = 6/5 , 270° < x < 360°

Answer 1

To find sin x/2, cos x/2, and tan x/2, we can use the Half-Angle Formulas. Given sec(x) = 6/5 and the range of x, we can find cos(x) and then substitute it into the formulas. The resulting values are sin x/2 = ±(1/2)√(1/3), cos x/2 = ±(1/2)√(11/3), and tan x/2 = (1/3)√(3/11).

Step-by-step explanation:

To find sin x/2, cos x/2, and tan x/2, we need to use the Half-Angle Formulas. In this case, we are given that sec(x) = 6/5 and that the angle x is in the range 270° to 360°.

The Half-Angle Formulas are:

1. sin(x/2) = ±√((1 - cos(x)) / 2)
2. cos(x/2) = ±√((1 + cos(x)) / 2)
3. tan(x/2) = sin(x/2) / cos(x/2)

Since sec(x) = 6/5, we can find cos(x) using the reciprocal identity: cos(x) = 1 / sec(x) = 5/6.

Now, substituting the value of cos(x) into the Half-Angle Formulas, we get:

1. sin(x/2) = ±√((1 - (5/6)) / 2) = ±√(1/12) = ±√(1/4) * √(1/3) = ±(1/2)√(1/3)
2. cos(x/2) = ±√((1 + (5/6)) / 2) = ±√(11/12) = ±√(1/4) * √(11/3) = ±(1/2)√(11/3)
3. tan(x/2) = sin(x/2) / cos(x/2) = ±(1/2)√(1/3) / ±(1/2)√(11/3) = (1/3)√(3/11)

Therefore, sin x/2 = ±(1/2)√(1/3), cos x/2 = ±(1/2)√(11/3), and tan x/2 = (1/3)√(3/11) (Note: the ± sign indicates that the values can be positive or negative depending on the quadrant).

Answer 2

To find sin x/2, cos x/2, and tan x/2 given sec(x) = 6/5 in the fourth quadrant, we utilize the half-angle formulas, realizing that cos(x) is positive and sin(x) negative in this quadrant. After simplifying the expressions and ensuring the signs match the quadrant, we acquire the desired trigonometric function values.

Step-by-step explanation:

To find sin x/2, cos x/2, and tan x/2 given that sec(x) = 6/5 and the angle x is in the fourth quadrant (270° < x < 360°), we must first understand the related trigonometric identities and implications of the angle being in the fourth quadrant.

Since sec(x) is the reciprocal of cos(x), we have cos(x) = 5/6. For angles in the fourth quadrant, cosine is positive, so we keep cos(x) = 5/6. Using the Pythagorean identity sin²(x) + cos²(x) = 1, we can find sin(x) to be negative in the fourth quadrant: sin(x) = -√(1 - cos²(x)) = -√(1 - (5/6)²).

With the half-angle formulas, we can find the half-angle trigonometric values:

• sin x/2 = √((1 - cos x)/2)
• cos x/2 = √((1 + cos x)/2)
• tan x/2 = sin x/2 / cos x/2

Therefore:

• sin x/2 = √((1 - 5/6)/2)
• cos x/2 = √((1 + 5/6)/2)
• tan x/2 = √((1 - 5/6)/(1 + 5/6))

The final task is to simplify these expressions taking care of the appropriate signs for the fourth quadrant