Which one of the following angles in standard position has the same reference angle as an angle of 215°?
A. 335°
B. 155°
C. 35°
D. 315°
Answer (2)
Find the reference angle of 215°, which is 215° − 180° = 35° .
Calculate the reference angles for each option: 335° (25°), 155° (25°), 35° (35°), and 315° (45°).
Compare the reference angles to find the matching angle.
The angle with the same reference angle as 215° is 35° .
Explanation
Find the reference angle of 215° First, we need to find the reference angle of 215°. The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. Since 215° is in the third quadrant, we subtract 180° from it to find the reference angle.
Calculate the reference angle The reference angle for 215° is calculated as follows: 215° − 180° = 35°
Calculate reference angles for each option Now, we need to find which of the given angles has the same reference angle of 35°. Let's calculate the reference angles for each option:
335°: Since 335° is in the fourth quadrant, we subtract it from 360°: 360° − 335° = 25°
155°: Since 155° is in the second quadrant, we subtract it from 180°: 180° − 155° = 25°
35°: Since 35° is in the first quadrant, its reference angle is itself: 35°
315°: Since 315° is in the fourth quadrant, we subtract it from 360°: 360° − 315° = 45°
Compare and find the matching angle Comparing the reference angles, we see that the angle 35° has the same reference angle as 215°.
Final Answer Therefore, the angle in standard position that has the same reference angle as an angle of 215° is 35°.
Examples
Understanding reference angles is crucial in trigonometry, especially when dealing with trigonometric functions of angles greater than 90 degrees. For example, if you're designing a solar panel and need to calculate the optimal angle for sunlight capture at a specific time of day, you might find that the sun's angle with respect to the horizon is 215 degrees. To simplify the trigonometric calculations for optimizing the panel's tilt, you would use the reference angle of 35 degrees, making the calculations easier while still achieving the correct result.
The reference angle of 215° is 35°. Among the options given, only angle 35° shares this same reference angle. Therefore, the correct answer is C. 35°.
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