Find the exact values of the six trigonometric functions for each of the following angles.draw the angles in standard position 1.300 degree
the reference angle is 60 degrees.
It's in QIV, so check to see which functions are positive or negative.
To find the exact values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for an angle, we need to first draw the angle in standard position.
For the angle of 300 degrees, we start by drawing the initial side of the angle pointing along the positive x-axis. Then, we rotate the terminal side of the angle counter-clockwise by 300 degrees.
Here is a visual representation of the angle in standard position:
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------------+------------> x-axis
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Now, let's find the exact values of the trigonometric functions for this angle.
1. Sine (sin): The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Since we are in standard position, we can create a right triangle by drawing a line segment from the terminal side of the angle perpendicular to the x-axis. In this case, the length of the side opposite (y-coordinate) will be 1, and the length of the hypotenuse will be the distance from the origin to the point on the terminal side of the angle.
To find the exact value of sin(300 degrees), we need to evaluate the sine function for this angle. Since sin(300 degrees) is equal to the y-coordinate of the point on the terminal side of the angle, we can conclude that sin(300 degrees) = 1.
2. Cosine (cos): The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In our right triangle, the length of the side adjacent (x-coordinate) will be determined by the distance from the origin to the point on the terminal side of the angle.
To find the exact value of cos(300 degrees), we need to evaluate the cosine function for this angle. Since cos(300 degrees) is equal to the x-coordinate of the point on the terminal side of the angle, we can conclude that cos(300 degrees) = 0.
3. Tangent (tan): The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. To find the exact value of tan(300 degrees), we can use the values we obtained for sin(300 degrees) and cos(300 degrees).
Since tan(300 degrees) = sin(300 degrees) / cos(300 degrees), we have tan(300 degrees) = 1 / 0. However, division by zero is undefined, indicating that the tangent of 300 degrees is undefined.
4. Cosecant (csc): The cosecant of an angle is defined as the reciprocal of the sine of the angle. To find the exact value of csc(300 degrees), we can use the value we obtained for sin(300 degrees).
Since csc(300 degrees) = 1 / sin(300 degrees), we have csc(300 degrees) = 1 / 1. Therefore, csc(300 degrees) = 1.
5. Secant (sec): The secant of an angle is defined as the reciprocal of the cosine of the angle. To find the exact value of sec(300 degrees), we can use the value we obtained for cos(300 degrees).
Since sec(300 degrees) = 1 / cos(300 degrees), we have sec(300 degrees) = 1 / 0, which is undefined.
6. Cotangent (cot): The cotangent of an angle is defined as the reciprocal of the tangent of the angle. To find the exact value of cot(300 degrees), we can use the values we obtained for tan(300 degrees).
Since cot(300 degrees) = 1 / tan(300 degrees), and tan(300 degrees) is undefined, cot(300 degrees) is also undefined.
Therefore, the exact values of the six trigonometric functions for an angle of 300 degrees are as follows:
sin(300 degrees) = 1
cos(300 degrees) = 0
tan(300 degrees) = undefined
csc(300 degrees) = 1
sec(300 degrees) = undefined
cot(300 degrees) = undefined.