evaluate the trigonometric function of the given quadrantal angle.
tan 1400°
cos 9π
cos 13π/2
sin (-17π)
subtract multiple of 360° (1400/360 = 3.8..
so
tan 1400°
= tan320° , 320 is in IV and 40° from the x-axis
= -tan40
= ....
cos 9π
= cos π , (took away 4 multiples of 2π)
= -1
cos 13π/2
= cos π/2 (took away 6π)
= 0
sin(-17π)
= sin(-17π + 18π) , ( I added 9 rotations)
= sin π
= 0
To evaluate the trigonometric functions of the given quadrantal angles, we need to use the unit circle and the periodicity of trigonometric functions.
1. tan 1400°:
First, we convert 1400° to the equivalent angle within one full revolution (360°) by dividing it by 360°:
1400° ÷ 360° = 3.8889 (approximately)
Since the value is greater than one complete revolution, we subtract the whole number part (3) and effectively retain just the fraction part (0.8889).
So, we are left with 0.8889 of a full revolution, which means we have to find the tangent of an angle of 0.8889 × 360° = 320°.
Looking at the unit circle, we find that the tangent function is positive in the second quadrant. Therefore, the tangent of 320° is the same as the tangent of the reference angle (180° - 320° = -140°), but with a positive sign:
tan 320° = tan (-140°) = tan 140°
Now you can evaluate the tangent function of 140° by using a calculator or a table lookup.
2. cos 9π:
Here, we have an angle in radians (9π). To evaluate it, we need to convert it to degrees.
Since π radians is equal to 180°, we can multiply 9π by 180°/π to get it in degrees:
9π × 180°/π = 1620°
Now, we have 1620°. However, this is greater than one full revolution, so we need to find its equivalent angle within one revolution by finding the remainder after dividing by 360°:
1620° ÷ 360° = 4.5
Since the value is greater than one complete revolution, we subtract the whole number part (4) and effectively retain just the fraction part (0.5).
So, we are left with 0.5 of a full revolution, which means we have to find the cosine of an angle of 0.5 × 360° = 180°.
Looking at the unit circle, we know that the cosine of 180° is -1.
Therefore, cos 9π = cos 1620° = cos 180° = -1.
3. cos 13π/2:
Here, we have 13π/2 as the angle in radians. To evaluate it, we calculate the angle in degrees.
We know that π radians is equal to 180°, so we can convert it as follows:
13π/2 × 180°/π = 13 × 90° = 1170°
Now, 1170° is greater than one full revolution, so we find its equivalent angle within one revolution by finding the remainder after dividing by 360°:
1170° ÷ 360° = 3.25
Since the value is greater than one complete revolution, we subtract the whole number part (3) and retain just the fraction part (0.25).
So, we are left with 0.25 of a full revolution, which means we have to find the cosine of an angle of 0.25 × 360° = 90°.
Looking at the unit circle, we know that the cosine of 90° is 0.
Therefore, cos 13π/2 = cos 1170° = cos 90° = 0.
4. sin (-17π):
Here, we have -17π as the angle in radians. To evaluate it, we calculate the angle in degrees.
Using the fact that π radians is equal to 180°:
-17π × 180°/π = -17 × 180° = -3060°
Now, -3060° is less than one full revolution, so we directly evaluate it.
Looking at the unit circle, we find that the sine of -3060° is equal to the sine of the reference angle (360° - 3060° = -2700°), but with a negative sign:
sin (-3060°) = sin (-2700°)
Now you can evaluate the sine function of -2700° by using a calculator or a table lookup.