Choose the three true statements from the following.
Options
A) The period of y =cosx is 90◦.
B) The period of y =sinx is ð radians.
C) The period of y =sin(2x) is four times the period of y =sinx.
D) The period of y =sin(2x) is half of the period of y =sinx.
E) The graph of y =sinx only cuts the x-axis only once.
F) The graph of y =cosx cuts the y-axis at y = −1.
G) sin(30◦) = sin(150◦)
H) cos(30◦)=cos(−30◦)
D G H
thanks
To determine the true statements, let's use the definitions and properties of trigonometric functions.
A) The period of y = cos(x) is 360° or 2π radians, not 90°. Therefore, statement A is false.
B) The period of y = sin(x) is also 360° or 2π radians. Therefore, statement B is true.
C) The period of y = sin(2x) can be found by dividing the period of y = sin(x) by the coefficient 2. Since the period of y = sin(x) is 360° or 2π radians, the period of y = sin(2x) is 180° or π radians. Four times the period of y = sin(x) would be 1440° or 8π radians. Therefore, statement C is false.
D) The period of y = sin(x) is 360° or 2π radians, and dividing that by 2 gives us 180° or π radians, which is indeed the period of y = sin(2x). Therefore, statement D is true.
E) The graph of y = sin(x) cuts the x-axis at multiple points, not just once. Therefore, statement E is false.
F) The graph of y = cos(x) cuts the y-axis at y = 1, not y = -1. Therefore, statement F is false.
G) The sine function has periodicity of 360° or 2π radians. Therefore, sin(30°) = sin(30° + 360°) = sin(390°), and sin(150°) = sin(150° + 360°) = sin(510°). Since the sine function repeats every 360°, sin(30°) = sin(150°). Therefore, statement G is true.
H) The cosine function is an even function, meaning cos(-x) = cos(x) for all x. Therefore, cos(30°) = cos(-30°). Therefore, statement H is true.
The three true statements are:
B) The period of y = sinx is ð radians.
D) The period of y = sin(2x) is half of the period of y = sinx.
H) cos(30◦)=cos(−30◦)