Trigonometric Functions: The Unit Circle Section 4.2

Objectives Identify a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Recognize the domain and range of sine and cosine functions Find the exact values of the trig functions at  /4 Use even and odd trig functions Recognize and use fundamental identities Periodic functions

Trigonometric Ratios The word trigonometry originates from two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements. A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent.

Trigonometric Ratios Only Apply to Right Triangles

In right triangles : The segment across from the right angle ( ) is labeled the hypotenuse “Hyp.”. The “angle of perspective” determines how to label the sides. Segment opposite from the Angle of Perspective( ) is labeled “Opp.” Segment adjacent to (next to) the Angle of Perspective ( ) is labeled “Adj.”. * The angle of Perspective is never the right angle. Hyp. Angle of Perspective Opp. Adj.

Labeling sides depends on the Angle of Perspective Angle of Perspective Hyp. Opp. Adj. Ifis the Angle of Perspective then …… * ”Opp.” means segment opposite from Angle of Perspective “Adj.” means segment adjacent from Angle of Perspective

If the Angle of Perspective is then Opp Hyp Adj then Opp Adj Hyp

The 3 Trigonometric Ratios The 3 ratios are Sine, Cosine and Tangent

Chief SohCahToa The Amazing Legend of…

Chief SohCahToa Once upon a time there was a wise old Native American Chief named Chief SohCahToa. He was named that due to an chance encounter with his coffee table in the middle of the night. He woke up hungry, got up and headed to the kitchen to get a snack. He did not turn on the light and in the darkness, stubbed his big toe on his coffee table…. Please share this story with Mr. Gustin for historical credibility.

Trigonometric Ratios To help you remember these trigonometric relationships, you can use the mnemonic device, SOH-CAH-TOA, where the first letter of each word of the trigonometric ratios is represented in the correct order. A C B b c a Sin A = Opposite side SOH Hypotenuse Cos A = Adjacent side CAH Hypotenuse Tan A = Opposite side TOA Adjacent side

SohCahToa

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are:  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle. The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp adj hyp θ

The Unit Circle Here we have a unit circle on the coordinate plane, with its center at the origin, and a radius of 1. The point on the circle is in quadrant I.

The Unit Circle Connect the origin to the point, and from that point drop a perpendicular to the x-axis. This creates a right triangle with hypotenuse of 1.

The Unit Circle x y 1  is the angle of rotation The length of its legs are the x- and y-coordinates of the chosen point. Applying the definitions of the trigonometric ratios to this triangle gives

The Unit Circle x y 1  is the angle of rotation The coordinates of the chosen point are the cosine and sine of the angle . –This provides a way to define functions sin(  ) and cos(  ) for all real numbers . –The other trigonometric functions can be defined from these.

Trigonometric Functions x y 1  is the angle of rotation These functions are reciprocals of each other.

Around the Circle As that point moves around the unit circle into quadrants I, II, III, and IV, the new definitions of the trigonometric functions still hold. I II III IV

The Unit Circle Completion One of the most useful tools in trigonometry is the unit circle. It is a circle, with radius 1 unit, that is on the x-y coordinate plane. 30º -60º -90º The hypotenuse for each triangle is 1 unit. 45º -45º -90º 30 º 60 º 1 45 º 1 In order to complete our unit circle with the missing coordinates, we must use the special right triangles below: cos sin The x-axis corresponds to the cosine function, and the y-axis corresponds to the sine function. 1

You first need to find the lengths of the other sides of each right triangle º 60 º 1 45 º 1 Use the Pythagorean Theorem to help find the sides. The two legs of a 45 – 45 – 90 triangle have the same length. Use the Pythagorean Theorem to find their sides. Find the remaining side.

Usefulness of Knowing Trigonometric Functions of Special Angles: 30 o, 45 o, 60 o The trigonometric function values derived from knowing the side ratios of the and triangles are “exact” numbers, not decimal approximations as could be obtained from using a calculator You will often be asked to find exact trig function values for angles other than 30 o, 45 o and 60 o angles that are somehow related to trig function values of these angles

Now, use the corresponding triangle to find the coordinates on the unit circle... (1, 0) sin cos (0, 1) (–1, 0) (0, –1) 30 º What are the coordinates of this point? (Use your triangle)

Now, use the corresponding triangle to find the coordinates on the unit circle... (1, 0) sin cos (0, 1) (–1, 0) (0, –1) What are the coordinates of this point? (Use your triangle) 45 º

You can use your special right triangles to find any of the points on the unit circle... (1, 0) sin cos (0, 1) (–1, 0) (0, –1) What are the coordinates of this point? (Use your triangle) 

You can use your special right triangles to find any of the points on the unit circle... (1, 0) sin cos (0, 1) (–1, 0) (0, –1) What are the coordinates of this point? (Use your triangle) Notice the coordinates in quadrants I and III.

Now we can complete the unit circle with the coordinates. (1, 0) sin cos (0, 1) (–1, 0) (0, –1)

Unit Circle (1, 0) (0, 1) (-1, 0) (0, -1) 0

Unit Circle (1, 0) (0, 1) (-1, 0) (0, -1) 30°

Unit Circle (1, 0) (0, 1) (-1, 0) (0, -1) 60°

Unit Circle (1, 0) (0, 1) (-1, 0) (0, -1) 45°

(1, 0) (0, 1) (–1, 0) (0, –1) Let’s look at a quick way to get the coordinates (in-class only)

You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly.

Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? 45° We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions. 45° 90° 0°0° 135° 180° 225° 270° 315° These are easy to memorize since they all have the same value with different signs depending on the quadrant.

Can you figure out what these angles would be in radians? The circle is 2  all the way around so half way is . The upper half is divided into 4 pieces so each piece is  /4. 45° 90° 0°0° 135° 180° 225° 270° 315°

Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division? 30° We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x. 30° 90° 0°0° 120° 180° 210° 270° 330° You'll need to memorize these too but you can see the pattern. 60° 150° 240° 300°

Can you figure out what the angles would be in radians? 30° It is still  halfway around the circle and the upper half is divided into 6 pieces so each piece is  /6. 30° 90° 0°0° 120° 180° 210° 270° 330° 60° 150° 240° 300° We'll see them all put together on the unit circle on the next screen.

(1,0) (0,1) (0,-1) (-1,0) We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions. 

(1,0) (0,1) (0,-1) (-1,0)  We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always 1 because it is a radius of the unit circle. Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

Finding Values of the Trigonometric Functions What are the coordinates?

Finding Values of the Trigonometric Functions What are the coordinates?

What is the domain? (domain means the “legal” things you can put in for  ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1  sin   1 (1, 0) (0, 1) (-1, 0) (0, -1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?)

Let’s think about the function f(  ) = cos  What is the domain? (domain means the “legal” things you can put in for  ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1  cos   1 (1, 0) (0, 1) (-1, 0) (0, -1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?)

ml Confused? Let’s look at an applet to unconfuse us. What is the domain of sine and cosine? All real numbers What is the range of sine and cosine?

Even and Odd Trig Functions

Remember negative angle means to go clockwise This is an even function. Think “same as”.

This is an odd function. Think “opposite”.

This is an odd function.

Using Even and Odd Functions to Find Values of Trig Functions

Reciprocal Identities

Quotient Identities

Using Quotient and Reciprocal Identities

Pythagorean Identities

Using a Pythagorean Identity

Periodic Functions

Periodic Properties of the Sine and Cosine Functions

Periodic Properties of the Tangent and Cotangent Functions

Sine and cosine are periodic with a period of 360 or 2 . We see that they repeat every  so the tangent’s period is . Let's label the unit circle with values of the tangent. (Remember this is just y/x)

Reciprocal functions have the same period. PERIODIC PROPERTIES sin(  + 2  ) = sin  csc(  + 2  ) = csc  cos(  + 2  ) = cos  sec(  + 2  ) = sec  tan(  +  ) = tan  cot(  +  ) = cot  This would have the same value as 1 (you can count around on unit circle or subtract the period twice.)

E XAMPLES : Evaluate the trigonometric function using its period as an aid

E XAMPLES : Evaluate the trigonometric function using its period as an aid