Find every angle θ between 0 and 360° for which the ratio of sin θ to cos θ is -3.00. (Round your answer to the nearest degree.)
tan T = sin T/ cos T
T = -71.56
tan is negative in quadrants 2 and 4
so
180 - 71.56 = 108
and
360 - 71.56 = 288
78
To find every angle θ between 0 and 360° for which the ratio of sin θ to cos θ is -3.00, we can use the trigonometric identity for tangent:
tan θ = sin θ / cos θ
Given that the ratio of sin θ to cos θ is -3.00, we have:
tan θ = -3.00
To find θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(-3.00)
Using a calculator, we find that arctan(-3.00) ≈ -71.57°.
This angle is in the second quadrant. To find the other angle in the fourth quadrant, we can use the fact that the tangent function has a period of 180°:
θ = -71.57° + 180° ≈ 108.43°
Therefore, the angles between 0 and 360° for which the ratio of sin θ to cos θ is -3.00 are approximately -71.57° and 108.43°.
To find each angle θ between 0° and 360° for which the ratio of sin θ to cos θ is -3.00, we can use trigonometric identities.
First, recall that sin θ / cos θ is equivalent to tan θ. Therefore, we need to find all angles θ for which tan θ equals -3.00.
Tan θ is defined as the ratio of sin θ to cos θ, so we can express tan θ as sin θ / cos θ = -3.00.
Next, we can use the inverse tangent function (also known as arctan or tan^(-1)) to find the angle θ that corresponds to a given value of tan θ.
Using a scientific calculator, we can find the angle θ that satisfies tan θ = -3.00. The inverse tangent of -3.00 is approximately -71.56°.
However, it is important to note that this value of -71.56° is not within the range of 0° to 360°. To find an angle within this range, we need to add or subtract integer multiples of 360°.
To find all possible angles, we can use the following formula: θ = arctan(-3.00) ± 360n, where n is an integer.
By substituting the value of arctan(-3.00) ≈ -71.56° and integer values of n, we can determine the angles within the desired range.
Here are a few examples:
- When n = 0, θ = -71.56° + 360(0) = -71.56°
- When n = 1, θ = -71.56° + 360(1) = 288.44°
- When n = 2, θ = -71.56° + 360(2) = 648.44°
We continue this process for other values of n until we find all the angles between 0° and 360° that satisfy the given condition.
Therefore, the angles θ between 0° and 360° for which the ratio of sin θ to cos θ equals -3.00 are approximately -71.56°, 288.44°, and 648.44° (rounded to the nearest degree).