Angle $D$ has a measure between $0^{\circ}$ and $360^{\circ}$ and is coterminal with a $-920^{\circ}$ angle. What is the measure of angle $D$?
Answer (2)
Express angle D using the concept of coterminal angles: D = − 92 0 ∘ + n ⋅ 36 0 ∘ .
Determine the integer value of n by solving the inequality 0 ∘ < − 92 0 ∘ + n ⋅ 36 0 ∘ < 36 0 ∘ , which gives n = 3 .
Calculate the measure of angle D : D = − 92 0 ∘ + 3 ⋅ 36 0 ∘ = 16 0 ∘ .
The measure of angle D is 16 0 ∘ .
Explanation
Express angle D using coterminality We are given that angle D is coterminal with a − 92 0 ∘ angle and that 0 ∘ < D < 36 0 ∘ . Coterminal angles differ by multiples of 36 0 ∘ . Therefore, we can express D as:
D = − 92 0 ∘ + n ⋅ 36 0 ∘ , where n is an integer.
Find the lower bound for n We need to find an integer n such that 0 ∘ < − 92 0 ∘ + n c d o t 36 0 ∘ < 36 0 ∘ .
First, let's find the lower bound for n :
0 < − 920 + 360 n 920 < 360 n \frac{920}{360} \approx 2.56"> n > 360 920 ≈ 2.56
Find the upper bound for n Next, let's find the upper bound for n :
− 920 + 360 n < 360 360 n < 1280 n < 360 1280 ≈ 3.56
Calculate the measure of angle D Since n must be an integer, the only possible value for n is 3 .
Now, we can find the measure of angle D :
D = − 92 0 ∘ + 3 ⋅ 36 0 ∘ = − 92 0 ∘ + 108 0 ∘ = 16 0 ∘
State the final answer Therefore, the measure of angle D is 16 0 ∘ .
Examples
Understanding coterminal angles is useful in fields like navigation and astronomy. For example, when tracking the movement of celestial bodies, angles are often measured beyond the standard 0-360 degree range. Converting these angles to their coterminal equivalents within 0-360 degrees simplifies calculations and makes it easier to determine the actual position of the object. This ensures accurate positioning and predictions in various applications.
The measure of angle D that is coterminal with − 92 0 ∘ is 16 0 ∘ . This is found by determining that n must be 3 to keep D in the range between 0 ∘ and 36 0 ∘ . The angle is calculated using the formula for coterminal angles.
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