Information about Trigonometry: Circular Functions

Discussions and examples on the unit circle and the circular functions

What is a UNIT CIRCLE? A unit circle is a circle whose radius is equal to 1 unit and its center is at the origin (0, 0). Substituting the coordinates of the center and the radius to the general equation of a circle would determine the equation of a unit circle.

Hence, the equation of the unit circle is (x - 0)2 + (y – 0)2 =1 or simply x2 + y2 = 1 The center (h, k) is (0, 0) and the radius (r) is 1. Since x² = (−x)² for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant.

Review: QUADRANTAL ANGLES Quadrantal angles are angles in standard position whose terminal ray lies along one of the axes. Examples are: 90 ° (π/2 radian), 180 ° (π radian), 270 ° ( 3π/2 radian) and 360 ° (2π radian) and their coterminal angles.

The Six Trigonometric functional identities in a unit circle Cos θ = x/r x Sin θ = y/r y Tan θ = y/x y/x Sec θ = r/x 1/x Csc θ = r/y 1/y Cot θ = x/y x/y x y r θ

Circular Functions of Quadrantal Angles Following the counterclockwise direction, the quadrantal angles dividing the unit circle are as follows: π/2 (90 °), π (180 °), 3(π)/2 (270 °), and 2π(360 °)If the direction point is clockwise, then the angles become negative: -π/2 (-90), -π(-180), -3π/2 (-270), - 2π(-360). (π) | 180 ° (π)/2 | 90 ° 3(π)/2 | 270 ° 2(π) | 360 °

As seen in this figure, The coordinates of π/2 is (0, 1) and lies on Quadrant I; the coordinates of π is (-1, 0) and lies on Quadrant II; the coordinates of 3π/2 is (0, -1) and lies on Quadrant III; while the coordinates of 2π is (0, -1) and lies on Quadrant IV. (π) | 180 ° (π)/2 | 90 ° 3(π)/2 | 270 ° 2(π) | 360 °

Radian Angle Coordinates Cos Sin Tan Sec Csc Cot π/2 90 ( 0, 1) 0 1 1 0 π 180 (-1, 0) -1 0 0 -1 3π/2 270 (0, -1) 0 -1 -1 0 2π 360 (1, 0) 1 0 0 1 To summarize, this table presents the quadrantal angles and their following coordinates, and trigonometric values. 8 8 8 8 8 8 8 8

Oral Exercise Find the value of the circular functions of the given quadrantal angles. 1. sin π 2 2. sin 3π 2 3. sin 8π 4 1 -1 0 4. sin 8π 8 1= = = 5. cos 5π 2 6. sec 10 π2 0 -1 = = =

Now that the coordinates of the quadrantal angles are defined, it is possible to identify the six trigonometric functions of each angles.

Review on Special Triangles Through the Pythagorean Theorem, the lengths of the sides of 45° - 45° and 30° - 60° - 90°right triangles are derived. x y r x2 + y2 = r2

7. cos 99 π99 8. sec 24 π2 -1 1 9. tan π 2 10. cot 3π 2 11. tan 8π 4 12. cot 8π 8 8 0 0 8 Oral Exercise Find the value of the circular functions of the given quadrantal angles. = = = = = =

Review on Special Triangles The length of the hypotenuse is equivalent to the length of the leg times square root of 2 in a 45 - 45 right triangle. a a a√2 45° 45° The length of the hypotenuse is equivalent to twice the length of the shorter leg (side opposite 30°), and the length of the longer leg (side opposite 60°) is equivalent to √3 times the shorter leg. a a√3 2a 30° 60°

The trigonometric functions of special angles would be determined with the aid of the unit circle. x y 1 θ Knowing the properties of these two special triangles will allow you to easily find the trigonometric functions of special angles, 30°, 45° and 60°.

Circular functions of multiples of 30° or π/6 x y 1 30°

Cos θ = x Sin θ = y Tan θ = y/x Sec θ = 1/x Csc θ = 1/y Cot θ = x/y Circular functions of multiples of 30° or π/6 Cos Sin Tan = = = √3 2 1 2 √3 3 Sec Csc Cot = = = 2√3 3 2 √3 Coordinates:

Circular functions of multiples of 150° or 5π/6

Circular functions of multiples of 210° or 7π/6

Circular functions of multiples of 330° or 11π/6

To summarize, this table presents the circular functions of π/3 and its multiples.

Circular functions of multiples of 45° or π/4 x y 1 45° 4 π 4 3π 4 7π 4 5π Since is in the 2nd Quadrant, its coordinates are Since is in the 3rd Quadrant, its coordinates are 4 3π 4 5π Since is in the 4th Quadrant, its coordinates are 4 7π ),( 2 2 2 2 ),( 2 2 2 2 − ),( 2 2 2 2 −− ),( 2 2 2 2 −

Cos θ = x Sin θ = y Tan θ = y/x Sec θ = 1/x Csc θ = 1/y Cot θ = x/y Circular functions of multiples of 45° or π/4 Cos Sin Tan = = = √2 2 √2 2 1 Sec Csc Cot = = = √2 Coordinates: ),( 2 2 2 2 4 π 4 π 4 π 4 π 4 π 4 π 1 √2

Cos Sin Tan = = = √2 2 √2 2 1 Sec Csc Cot = = = √2 Coordinates: ) 2 2 , 2 2 (− 4 3π 1 √24 3π 4 3π 4 3π 4 3π 4 3π - - Circular functions of multiples of 135° or 3π/4 - -

Cos Sin Tan = = = √2 2 √2 2 1 Sec Csc Cot = = = √2 Coordinates: ) 2 2 , 2 2 ( −− 4 3π 1 √24 3π 4 3π 4 3π 4 3π 4 3π - - Circular functions of multiples of 225° or 5π/4 - -

Cos Sin Tan = = = √2 2 √2 2 1 Sec Csc Cot = = = √2 Coordinates: ) 2 2 , 2 2 ( − 4 3π 1 √24 3π 4 3π 4 3π 4 3π 4 3π - - Circular functions of multiples of 315° or 7π/4 - -

Circular functions of multiples of 60° or π/3 x y 1 60° 3 π 3 2π 3 5π 3 4π Since is in the 2nd Quadrant, its coordinates are Since is in the 3rd Quadrant, its coordinates are Since is in the 4th Quadrant, its coordinates are ),( 2 3 2 1 3 2π 3 4π 3 5π ),( 2 3 2 1 − ),( 2 3 2 1 − ),( 2 3 2 1 −−

Cos θ = x Sin θ = y Tan θ = y/x Sec θ = 1/x Csc θ = 1/y Cot θ = x/y Circular functions of multiples of 60° or π/3 Cos Sin Tan = = = 1 2 Sec Csc Cot = = = 2 Coordinates: 3 π 3 π 3 π 3 π 3 π 3 π ),( 2 3 2 1 √3 2 √3 2√3 3 √3 3

Exercise #1 – Part A Directions: Write True if the statement is correct; otherwise, changed the underlined word. Write your answers in a whole sheet of paper. You only have 5 minutes to answer the following. _________1. Quadrantal angles are angles whose terminal rays lies in one of the axes. _________2. (π)/2 lies in the positive x-axis. _________3. Quadrantal real numbers are numbers whose starting and terminal points lies on one of the axes. _________4. The value of cos (π) is 0. _________5. 3(π)/2 is equivalent to 360 ° . Let us Check! 1. True 2. Negative 3. Arc lengths 4. -1 5. 2(π)

Exercise #1 – Part B Directions: Identify the values being asked in the following. Write your answers in a one whole sheet of paper. You only have 15 minutes to answer the following. ________1. sin(π) ________ 2. cot (2π) ________3. sec (5π/2) ________4. Csc (3 π/2) ________5. Tan (π/2) Let us Check! 1. 0 2. Undefined 3. Undefined 4. -1 5. Undefined ________6. Cos (3π) ________7. tan (6π) ________8. Sin (11/2) ________9. Sec (7π/2) ________10. Cot (4π) 6. -1 7. 0 8. -1 9. -1 10. undefined

Exercises 1. sin45 + cot210 = 2. sec30+ tan 135= 3. csc630 – cot210 + tan45= 4. sin240+ cos315= 5. sin90+ cos60= Let Us Check! 1. 2. 3. 3 2 2 + 2 322 + 1 3 32 + 3 332 + 131 +−− 3− 4. 5. 2 2 2 3 +− 2 23 +− 2 1 1+ 2 3

Exercises 1. sin60= 2. cot30= 3. tan150= 4. sec450= 5. csc120= 6. sin90= 1. sin60= 2. cot30= 3. tan150= 4. sec450= 5. csc120= 6. sin90= Let Us Check! 1. sin60 = 2. cot30 = 3. tan150 = 4. sec450 = 5. csc120 = 6. sin90 = √3 2 √3 √3 3 - 0 2√3 3 1

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