Find all angles θ between 0° and 180° satisfying the given equation. Round your answer to one decimal place.
cos(θ) = − 4/5
cosθ is negative in QII and, QIII
cos 36.9° = 4/5
so use that as your reference angle, and you get
180-36.9°
To find all angles θ between 0° and 180° satisfying the equation cos(θ) = -4/5, you can use the inverse cosine function (also known as arccosine or cos^(-1)).
Step 1: Apply the inverse cosine function to both sides of the equation: arccos(cos(θ)) = arccos(-4/5).
Step 2: Simplify the left side: θ = arccos(-4/5).
Step 3: Evaluate the right side using a scientific calculator or table of trigonometric values. arccos(-4/5) is approximately equal to 143.1°.
Step 4: Since we are looking for all angles between 0° and 180°, we need to consider the reference angle (the angle within the first quadrant) and its corresponding angle in the second quadrant.
The reference angle, since we are dealing with a cosine function, can be found by taking the inverse cosine of the absolute value of the given ratio: arccos(|-4/5|) = arccos(4/5). This is approximately equal to 36.9°.
Step 5: To find the first solution in the second quadrant, subtract the reference angle from 180°: 180° - 36.9° = 143.1°.
Therefore, the angles θ that satisfy the equation cos(θ) = -4/5 between 0° and 180° are approximately 36.9° and 143.1° (rounded to one decimal place).
To find all angles θ between 0° and 180° satisfying the equation cos(θ) = -4/5, we can use the inverse cosine function (cos^(-1)).
The inverse cosine function returns the angle whose cosine is equal to the given value.
Let's calculate the value using the inverse cosine function:
θ = cos^(-1)(-4/5)
θ ≈ 139.2°
Therefore, the angle θ ≈ 139.2° satisfies the equation cos(θ) = -4/5 between 0° and 180°.