15.1 Trigonometric Functions, the Unit Circle and Radians
This page defines the six trigonometric functions from the unit circle, works out their values at the standard angles, converts between degrees and radians, and begins graphing them. The exercises at the end evaluate a sinusoid and solve some equations.
Defining the trigonometric functions
Definition 15.1.1. The cosine() of an angle is the -coordinate of the terminal point of . The sine() is the -coordinate of that same terminal point. The tangent() is the ratio of the -coordinate to the -coordinate (i.e. ).
Definition 15.1.2. The secant is , the cosecant is , and the cotangent is .
Problem 15.1.3. The side lengths of a -- triangle and a -- triangle, each with hypotenuse .
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Problem 15.1.4. Cosine, sine and tangent of ,,,, and .
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Problem 15.1.5. Let be a right triangle such that . Now let be coordinates on the unit circle such that and , so the point is .
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So then :
The -coordinate is and the -coordinate is . The same follows for all other angles and ratios.
Problem 15.1.6. Where sine and tangent are positive.
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(a) Sine is positive in quadrants I and II.
(b) Tangent is positive in quadrants I and III.
Problem 15.1.7. Values at angles outside the first quadrant.
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(a)
(b)
(c)
(d)
(e)
(f)
Problem 15.1.8. Values at angles beyond and negative angles.
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(a)
(b)
(c)
(d)
Problem 15.1.9. Given and , show .
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Note by the Pythagorean theorem we have . Also note that since and , and is equal to the distance from the origin in the -direction and the same for with the -direction, then and for the right triangle in the unit circle. And since the hypotenuse is , by the Pythagorean theorem we have
Problem 15.1.10. Ranges of sine, tangent and secant.
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(a) where is .
(b) with is .
(c) with is .
Radians
Note. An angle in degrees is radians, where
Problem 15.1.11.,, and in radians.
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Problem 15.1.12.,, and radian in degrees.
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Problem 15.1.13. Trigonometric functions of angles in radians.
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Problem 15.1.14.Show solution
Negative.
Problem 15.1.15. The domain of .
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Graphing trigonometric functions
Problem 15.1.16. for .
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We show that is periodic.
Definition 15.1.17. A function is periodic if there exists such that for all such that and are defined.
Proof. Let . Note that , and since (as angles), for all . The same follows for all trigonometric functions: period of .∎
Graph of , with dotted:
Exercises
Sinusoids
Exercise 15.1.18. Evaluate at , and .
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Trigonometric equations
Note. Rule of note: when arc… (it is at a peak), it will only have one solution (per period).