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?
your answers are correct, (but I have a feeling you used your calculator and then decided what radical your answer was)
Some students find it easier to "think" in degrees than radians
10pi/3 radians = 600º
you have to change that to a co-terminal angle, that is, an angle in the first rotation landing in the same position.
To do that subtract multiples of 360 until you have an answer less than 360
600-360 = 240
Now ask yourself, "Where is 240º ?"
- in the third quadrant.
In III, the sine is -, the cosine is +, and the tangent is +
Now ask yourself, "What is the angle in standard position equal to 240º ?"
- that is, how far to the x-axis ? -- 60º
You should know the sine, cosine and tangents of 60º
Using the proper sines for III
sin 10pi/3 = sin 600º = sin 240º = -sin 60º = -√3/2
etc.
Practice with a few, after a while this is very quick and easy.
To evaluate the sine, cosine, and tangent of angles without a calculator, you can use the unit circle and the properties of the trigonometric functions.
1) For angle a) 10π/3:
To determine the sign of the trigonometric functions, consider the quadrant in which the angle falls. In this case, 10π/3 is in the third quadrant, where both the x-coordinate and y-coordinate are negative.
Sin = -sqrt(3)/2: In the third quadrant, the sine is negative. The value of sin(10π/3) is -sqrt(3)/2.
Cos = -1/2: In the third quadrant, the cosine is negative. The value of cos(10π/3) is -1/2.
Tan = sqrt(3): In the third quadrant, the tangent is positive. The value of tan(10π/3) is sqrt(3).
So, your answers for angle a) are correct.
2) For angle b) 17π/3:
To determine the sign of the trigonometric functions, consider the quadrant in which the angle falls. In this case, 17π/3 is in the fifth quadrant, where the x-coordinate is positive, and the y-coordinate is negative.
Sin = -sqrt(3)/2: In the fifth quadrant, the sine is negative. The value of sin(17π/3) is -sqrt(3)/2.
Cos = 1/2: In the fifth quadrant, the cosine is positive. The value of cos(17π/3) is 1/2.
Tan = -sqrt(3): In the fifth quadrant, the tangent is negative. The value of tan(17π/3) is -sqrt(3).
So, your answers for angle b) are correct.
Remember, when determining the sign of the trigonometric functions, you can visualize the unit circle and the quadrant in which the angle falls. Based on the coordinates of the point on the unit circle corresponding to the angle, you can determine whether the trigonometric function is positive or negative.