Let � be an angle in standard position and the point
(a, b) be the point of intersection of the terminal
side of � with the unit circle. State the unit circle
definitions of the six trigonometric functions.
cos� = sec� =
sin� = csc� =
tan� = cot� =
If Ø is your angle, and since
r=√(a^2 + b^2)
sinØ = b/r
cscØ = r/b
cosØ = a/r
secØ = r/a
tanØ = b/a
cotØ = a/b
These definitions should be in your textbook and are probably defined in terms of x, y and r.
You should memorize them
Sure! The unit circle definitions of the six trigonometric functions are as follows:
1. Cosine (cos θ): The cosine of θ is equal to the x-coordinate of the point (a, b) on the unit circle. So, cos θ = a.
2. Secant (sec θ): The secant of θ is equal to the reciprocal of the cosine of θ. So, sec θ = 1/cos θ.
3. Sine (sin θ): The sine of θ is equal to the y-coordinate of the point (a, b) on the unit circle. So, sin θ = b.
4. Cosecant (csc θ): The cosecant of θ is equal to the reciprocal of the sine of θ. So, csc θ = 1/sin θ.
5. Tangent (tan θ): The tangent of θ is equal to the ratio of the sine of θ to the cosine of θ. So, tan θ = sin θ / cos θ.
6. Cotangent (cot θ): The cotangent of θ is equal to the reciprocal of the tangent of θ. So, cot θ = 1/tan θ.
These definitions are based on the coordinates of the point (a, b) on the unit circle and provide relationships between the trigonometric functions.
To determine the unit circle definitions of the six trigonometric functions, we need to relate the coordinates of the point (a, b) on the unit circle to these functions. Consider the right triangle formed by the point (a, b), where the hypotenuse is the radius of the unit circle (which is always 1), and the other two sides are the horizontal distance (a) and vertical distance (b) from the origin to the point on the unit circle.
The definitions of the six trigonometric functions are as follows:
1. cosine (cos): It is defined as the ratio of the adjacent side (a) to the hypotenuse (1).
Therefore, cos� = a/1 = a.
2. secant (sec): It is defined as the reciprocal of the cosine, so sec� = 1/cos�.
Substituting the value of cos�, we get sec� = 1/a.
3. sine (sin): It is defined as the ratio of the opposite side (b) to the hypotenuse (1).
Therefore, sin� = b/1 = b.
4. cosecant (csc): It is defined as the reciprocal of the sine, so csc� = 1/sin�.
Substituting the value of sin�, we get csc� = 1/b.
5. tangent (tan): It is defined as the ratio of the opposite side (b) to the adjacent side (a).
Therefore, tan� = b/a.
6. cotangent (cot): It is defined as the reciprocal of the tangent, so cot� = 1/tan�.
Substituting the value of tan�, we get cot� = a/b.
In summary, the unit circle definitions of the six trigonometric functions are:
cos� = a
sec� = 1/a
sin� = b
csc� = 1/b
tan� = b/a
cot� = a/b