ACC Module #3 Unit 3.6

1. DemingEarly College High SchoolUnit 3.0 Advanced Algebra and Functions (AAF) 3.6 Trigonometry

2. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry Trigonometric functions are built out of two basic functions, the sine and cosine, written as and respectively. Note the similar to logarithms, it is customary to drop the parentheses as long as the result is not confusing. The sine and cosine are defined using the unit circle. If is the angle going counterclockwise around the origin from the x-axis, then the point on the unit circle in that direction will have the coordinates .

3. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry Since the angle returns to the start every radians (or 360 degrees), the graph of these functions will be periodic, with a period of . This means that the graph repeats itself as one moves along the x-axis because . Cosine works similarly. From the unit circle definition, the sine function starts at 0 when . It grows to 1 as grows to , and then back to 0 at . Then it decreases to -1 as grows to and back up to 0 at

4. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry The graph of the cosine is similar. The cosine will start at 1, decreasing to 0 at and continue to decrease to -1 at . Then, it grows to 0 as grows to and back up to 1 at .

5. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry Another trigonometric function, which is frequently used, is the tangent function. This is defined as the following equation: The tangent function is a period of rather than because the sine and cosine functions have the same absolute values after a change in the angle of , but flip their signs. Since the tangent is a ratio of the two functions, the changes in signs cancel. The tangent function will be zero when the sin is zero, and it will have a vertical asymptote whenever cosine is zero.

6. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry Three other trigonometric functions are some times useful. The are the reciprocal trigonometric functions, so named because they are just the reciprocals of sine, cosine, and tangent. They are: cosecant, defined as secant, defined as cotangent, defined as . Note that from the definition of tangent, .

7. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry In addition, there are three identities that relate the trigonometric functions to one another: On the following page is a list of commonly needed values for trigonometric functions, given in radians, for the first quadrant.

8. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry

9. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry To find the trigonometric values in other quadrants, complementary angles can be used. The complementary angle is the smallest angle between the x-axis and the given angle. Once a complementary angle is know, the following rule is used: The correct sign is used based on the functions sine and cosine are given by the x and y coordinates on the unit circle. Sine will be positive in quadrants I and II and negative in quadrants III and IV. Cosine will be positive in quadrants I and IV and negative in quadrants II and III. Tangent will be positive in quadrants I and III and negative in II and IV. For an angle with complementary angle x, the absolute value of a trigonometry function is evaluated at is the same as the absolute value when evaluated at x.

10. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry The signs of the reciprocal functions will be the same as the sign of the function of which they are a reciprocal. For example, find . First, the complementary angle must be found. This angle is in the II quadrant, and the angle between it and the x-axis is . But, we know . Since this is in the II quadrant, sine take on positive values (the y coordinate is positive in the II quadrant. Therefore, . In addition to the six trigonometric functions defined previously, there are inverses for these functions. However, since the trigonometric functions are not one-to-one, one can only construct inverses for them on a restricted domain.

11. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry Usually, the domain chosen will be for cosine and for sine. The inverse for tangent can use either of these domains. The inverse functions for the trigonometric functions are also called arc functions. In addition to being written with a -1 in the exponent to denote that the function is an inverse, they will sometimes be written with an “a” or “arc” in front of the function name, so . When solving equations that involve trigonometric functions, there are often multiple solutions. For example, cab be simplified to . This also has solutions , but in addition because of the periodicity, any integer multiple of can also be added to these solutions to find another solution.

12. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry The full set of solutions is for all integer values of k. It is very important to remember to find all possible solutions when dealing with equations that involve trigonometric functions. The name trigonometric comes from the fact that these functions play an important role in the geometry of triangles, particularly right triangles.

13. Unit 3.0 Advanced Algebra and Functions (AAF)3.6 Trigonometry Finally, consider this right triangle: Remember also the angles of a triangles must add up to radians (180 degrees)!