evaluate the trigonometric function of the given quadrantal angle.
tan 1440°
Evaluate without using ratios in reference triangle.
sec 3π/4
tan 1440° = tan 1440-4(360) = tan 1440-1440 = tan 0° = 0
sec 3π/4 = -sec π/4 = -√2
To evaluate the trigonometric function tan 1440° without using ratios in a reference triangle, first, we need to simplify the angle to its equivalent angle within one revolution (360°).
1440° is equivalent to (4 × 360°) + 0°, which means it completes four full revolutions. Therefore, the angle 1440° is the same as 0°, since starting from 0° and making four full revolutions brings us back to the starting position.
Now, we can evaluate the tan of 0°:
tan 0° = sin 0° / cos 0°
The sine of 0° is 0, and the cosine of 0° is 1.
Therefore, tan 0° = 0 / 1 = 0.
Hence, tan 1440° = 0.
To evaluate the trigonometric function of a quadrantal angle, we need to understand that quadrantal angles fall directly on the axes in the coordinate plane (90°, 180°, 270°, 360°, etc.).
For the given problem, tan 1440°, we can simplify it by using the periodic nature of the tangent function. Since 360° is a full rotation, we can subtract 360° from 1440° to find an equivalent angle within one full rotation.
1440° - 360° = 1080°
Now, we can evaluate tan 1080°. To do this, recall that tangent is equal to the ratio of the sine of an angle to the cosine of the same angle.
tan θ = sin θ / cos θ
In this case, tan 1080° can be written as sin 1080° / cos 1080°.
However, because we are dealing with a quadrantal angle, both the sine and cosine values will be equal to zero. Therefore, tan 1080° is undefined, as division by zero is not possible.
Moving on to the second problem, evaluating sec 3π/4 without using ratios in a reference triangle, we can first think about finding the secant function using its reciprocal relationship with cosine.
sec θ = 1/cos θ
In this case, we need to evaluate sec 3π/4. To do this, we can directly find the cosine of 3π/4.
Recall that cosine is negative in the 2nd and 3rd quadrants of the unit circle. Since 3π/4 falls into the 2nd quadrant, the cosine value of 3π/4 will be negative.
By evaluating the exact value of cosine at 3π/4, which is -1/√2, we can calculate the secant of 3π/4 as follows:
sec 3π/4 = 1 / cos 3π/4 = 1 / (-1/√2) = -√2