1) Evaluate (if possible) the sine, cosine, and tangent of the angles without a calculator.
a) 10pi/3
Sin = -sqrt 3/2
Cos = -1/2
Tan = sqrt 3
Are these correct ? I do not understand when to make them negative.
b) 17pi/3
Sin = -sqrt 3/2
Cos = 1/2
Tan = - sqrt 3
Are these correct?
a) For angle 10π/3:
Sin = -√3/2
Cos = -1/2
Tan = √3
Yes, these values are correct. In determining whether to make them negative, you need to consider the quadrant in which the angle lies. Starting from the positive x-axis and moving counterclockwise, the quadrants are labeled as follows:
Quadrant I: sin, cos, and tan are all positive.
Quadrant II: sin is positive, cos is negative, tan is negative.
Quadrant III: sin, cos is negative, tan is positive.
Quadrant IV: sin is negative, cos is positive, tan is negative.
In the case of 10π/3, it lies in Quadrant IV. Therefore, sin is negative, cos is negative, and tan is positive.
b) For angle 17π/3:
Sin = -√3/2
Cos = 1/2
Tan = -√3
Yes, these values are correct. Angle 17π/3 also lies in Quadrant IV, so sin is negative, cos is positive, and tan is negative.
To evaluate sine, cosine, and tangent of an angle without using a calculator, you can refer to the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. The values of sine and cosine for the angles on the unit circle are typically memorized.
a) Evaluating for angle 10pi/3:
Step 1: Determine the reference angle:
The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. To find the reference angle, subtract or add 2π (360 degrees) to the given angle until it falls within the range of 0 to 2π (0 to 360 degrees). In this case, 10pi/3 is equivalent to (6pi + 4pi)/3, which is equivalent to 4pi/3. Therefore, the reference angle is 4pi/3.
Step 2: Determine the sign of the trigonometric functions:
In the first quadrant (0 to π/2), both sine and cosine are positive. In the second quadrant (π/2 to π), only sine is positive. In the third quadrant (π to 3π/2), only tangent is positive. In the fourth quadrant (3π/2 to 2π), only cosine is positive.
Step 3: Evaluate the trigonometric functions using the reference angle:
For angle 4pi/3, the reference angle is in the third quadrant where only tangent is positive.
- Sin = -sin(4pi/3) = -sqrt(3)/2 (negative because sine is positive in the second quadrant)
- Cos = -cos(4pi/3) = -1/2 (negative because cosine is positive in the second quadrant)
- Tan = tan(4pi/3) = sqrt(3) (positive because tangent is positive in the third quadrant)
Therefore, the correct values for angle 10pi/3 are:
Sin = -sqrt(3)/2
Cos = -1/2
Tan = sqrt(3)
b) Evaluating for angle 17pi/3:
Step 1: Determine the reference angle:
Following the same process as above, 17pi/3 is equivalent to (6pi + 5pi)/3, which is equivalent to 5pi/3. Therefore, the reference angle is 5pi/3.
Step 2: Determine the sign of the trigonometric functions:
In the first quadrant (0 to π/2), both sine and cosine are positive. In the second quadrant (π/2 to π), only sine is positive. In the third quadrant (π to 3π/2), only tangent is positive. In the fourth quadrant (3π/2 to 2π), only cosine is positive.
Step 3: Evaluate the trigonometric functions using the reference angle:
For angle 5pi/3, the reference angle is in the third quadrant where only tangent is positive.
- Sin = -sin(5pi/3) = -sqrt(3)/2 (negative because sine is positive in the second quadrant)
- Cos = cos(5pi/3) = 1/2 (positive because cosine is positive in the fourth quadrant)
- Tan = tan(5pi/3) = -sqrt(3) (positive because tangent is positive in the third quadrant)
Therefore, the correct values for angle 17pi/3 are:
Sin = -sqrt(3)/2
Cos = 1/2
Tan = -sqrt(3)
Both sets of values you provided are correct. Remember to consider the sign for each trigonometric function based on the quadrant where the angle falls.