Determine if true or false for each question:
For an angle A in standard position, if sinA = cosA then the terminal arm of the angle lies in quadrant II or IV
For an angle A in standard position, if sinA = -cosA then the terminal arm of the angle lies in quadrant I or III
the sine is positive in both I and II
the cosine is positive in both I and IV
the sine is negative in both III and IV
the cosines is negative in both II and III
so sin A = cos A in only I and III , so your first statement is false.
Repeat the same argument for the 2nd question.
Okay so if I apply that logic to the second statement, the sine will stay the same but the quadrants for the negatives and positives of cosine will swap.
the sine is positive in both I and II
the -cosine is positive in both II and III
the sine is negative in both III and IV
the -cosines is negative in both I and IV
That means sinA = -cosA is only in II and IV, making the second statement also false.
Right
To determine if these statements are true or false, we can use the definitions and properties of the trigonometric functions.
1. For an angle A in standard position, if sinA = cosA, then the terminal arm of the angle lies in quadrant II or IV.
To verify this statement, we need to recall the definitions of sine and cosine in the unit circle. The sine of an angle is the y-coordinate of the point where the terminal arm intersects the unit circle, and the cosine of an angle is the x-coordinate of the same point.
In quadrant II, both the sine and cosine are positive. In quadrant IV, the sine is negative and the cosine is positive. Therefore, for sinA = cosA to be true, angle A must lie in both quadrant II and quadrant IV.
2. For an angle A in standard position, if sinA = -cosA, then the terminal arm of the angle lies in quadrant I or III.
Again, we refer to the definitions of sine and cosine. In quadrant I, both the sine and cosine are positive, while in quadrant III, both the sine and cosine are negative. Therefore, for sinA = -cosA to be true, angle A must lie in both quadrant I and quadrant III.
Therefore, the answers to the questions are as follows:
1. False: If sinA = cosA, the terminal arm of the angle lies in quadrants II and IV. (Not just one or the other)
2. False: If sinA = -cosA, the terminal arm of the angle lies in quadrants I and III. (Not just one or the other)