Given: sec θ = –6.4, 0˚ θ 360˚ , solve for θ.
To solve for θ, we need to find the angle whose secant value is -6.4.
The secant function (sec) is the reciprocal of the cosine function, so we can write it as:
sec θ = 1/cos θ
Therefore, we have:
1/cos θ = -6.4
To solve this equation for θ, we need to find the angle whose cosine value is the reciprocal of -6.4, which is -1/6.4 or -0.15625.
We can use the inverse cosine function (arccos) to find the angle. However, the arccos function is generally defined for outputs in the range [0, π], or [0°, 180°].
Since the given range for θ is [0°, 360°], we need to consider both positive and negative angles.
Let's find the arccos of -0.15625 using a calculator:
arccos(-0.15625) ≈ 1.7294 radians ≈ 99.2 degrees (rounded to one decimal place)
We have found one angle, but we need to consider both positive and negative angles. To find the second angle, we subtract the first angle from 360 degrees:
360° - 99.2° ≈ 260.8° (rounded to one decimal place)
Therefore, the two angles that satisfy the equation sec θ = -6.4 are approximately 99.2° and 260.8°.