Unit Circle Trigonometry Text 1

UNIT CIRCLE TRIGONOMETRY The Unit Circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). The equation of this circle is x 2 + y 2 = 1 . A diagram of the unit circle is shown below: y x2 + y2 = 1 1 x -2 -1 1 2 -1 -2 We have previously applied trigonometry to triangles that were drawn with no reference to any coordinate system. Because the radius of the unit circle is 1, we will see that it provides a convenient framework within which we can apply trigonometry to the coordinate plane. Drawing Angles in Standard Position We will first learn how angles are drawn within the coordinate plane. An angle is said to be in standard position if the vertex of the angle is at (0, 0) and the initial side of the angle lies along the positive x-axis. If the angle measure is positive, then the angle has been created by a counterclockwise rotation from the initial to the terminal side. If the angle measure is negative, then the angle has been created by a clockwise rotation from the initial to the terminal side. θ in standard position, where θ is positive: y θ in standard position, where θ is negative: y Terminal side θ Initial side Initial side x θ x Terminal side Unit Circle Trigonometry Drawing Angles in Standard Position Examples The following angles are drawn in standard position: y 1. θ = 40D 2. θ = 160D θ y θ x x y 3. θ = −320 D Notice that the terminal sides in examples 1 and 3 are in the same position, but they do not represent the same angle (because the amount and direction of the rotation in each is different). Such angles are said to be coterminal. x θ Exercises Sketch each of the following angles in standard position. (Do not use a protractor; just draw a brief sketch.) 1. θ = 120D 2. θ = − 45D y 3. θ = −130D y y x 4. θ = 270D x θ = −90D y Unit Circle Trigonometry 6. θ = 750D y x x y x Drawing Angles in Standard Position x Labeling Special Angles on the Unit Circle We are going to deal primarily with special angles around the unit circle, namely the multiples of 30o, 45o, 60o, and 90o. All angles throughout this unit will be drawn in standard position. First, we will draw a unit circle and label the angles that are multiples of 90o. These angles, known as quadrantal angles, have their terminal side on either the x-axis or the yaxis. (We have limited our diagram to the quadrantal angles from 0o to 360o.) Multiples of 90o: y 90o 1 180o 1 -1 0o 360o x Note: The 0o angle is said to be coterminal with the 360o angle. (Coterminal angles are angles drawn in standard position that share a terminal side.) -1 270o Next, we will repeat the same process for multiples of 30o, 45o, and 60o. (Notice that there is a great deal of overlap between the diagrams.) Multiples of 30o: y 120o 90o Multiples of 45o: y 90o 60o 1 135 30o 150o 180o 1 -1 210o 0o 360o x o 1 45o 180o 1 -1 0o 360o x 330o 240o -1 270 225o 300o o 315o -1 270 o Multiples of 60o: y 120o 1 60o 180o 1 -1 240o -1 Unit Circle Trigonometry 0o 360o x 300o Labeling Special Angles on the Unit Circle Putting it all together, we obtain the following unit circle with all special angles labeled: y 90o 120o 1 60o 135o 45o 150o 30o 0o 180o 1 -1 210o 360o x 330o 315o 225o -1 240o 300 o 270o Coordinates of Quadrantal Angles and First Quadrant Angles We want to find the coordinates of the points where the terminal side of each of the quadrantal angles intersects the unit circle. Since the unit circle has radius 1, these coordinates are easy to identify; they are listed in the table below. y 90o (0, 1) 1 0o (-1, 0) 180o 1 -1 -1 270o (1, 0) 360o x Angle 0o 90o 180o 270o 360o Coordinates (1, 0) (0, 1) (-1, 0) (0, -1) (1, 0) (0, -1) We will now look at the first quadrant and find the coordinates where the terminal side of the 30o, 45o, and 60o angles intersects the unit circle. Unit Circle Trigonometry Coordinates of Quadrantal Angles and First Quadrant Special Angles First, we will draw a right triangle that is based on a 30o reference angle. (When an angle is drawn in standard position, its reference angle is the positive acute angle measured from the x-axis to the angle’s terminal side. The concept of a reference angle is crucial when working with angles in other quadrants and will be discussed in detail later in this unit.) y 1 30o 1 -1 x -1 Notice that the above triangle is a 30o-60o -90o triangle. Since the radius of the unit circle is 1, the hypotenuse of the triangle has length 1. Let us call the horizontal side of the triangle x, and the vertical side of the triangle y, as shown below. (Only the first quadrant is shown, since the triangle is located in the first quadrant.) y 1 ( x, y ) 1 60o y 30o x 1 x We want to find the values of x and y, so that we can ultimately find the coordinates of the point ( x, y ) where the terminal side of the 30o angle intersects the unit circle. Recall our theorem about 30o-60o-90o triangles: In a 30o-60o-90o triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. Unit Circle Trigonometry Coordinates of Quadrantal Angles and First Quadrant Special Angles Since the length of the hypotenuse is 1 and it is twice the length of the shorter leg, y, we can say that y = 12 . Since the longer leg, x, is 3 times the length of the shorter leg, we can say that x = 1 2 3 , or equivalently, x = 3 2 . Based on the values of the sides of the triangle, we now know the coordinates of the point ( x, y ) where the terminal side of the 30o angle intersects the unit circle. This is the point ( 3 2 ) , 12 , as shown below. y 1 ( 1 30 60o o 3 2 ) 1 2 1 3 2 , 12 x We will now repeat this process for a 60o reference angle. We first draw a right triangle that is based on a 60o reference angle, as shown below. y 1 ( x, y ) 1 y 60o x 1 x We again want to find the values of x and y. The triangle is a 30o-60o -90o triangle. Since the length of the hypotenuse is 1 and it is twice the length of the shorter leg, x, we can say that x = 12 . Since the longer leg, y, is 3 times the length of the shorter leg, we can say that y = 1 2 3 , or equivalently, y = Unit Circle Trigonometry 3 2 . Coordinates of Quadrantal Angles and First Quadrant Special Angles Based on the values of the sides of the triangle, we now know the coordinates of the point ( x, y ) where the terminal side of the 60o angle intersects the unit circle. This is the point ( 1 2 , 3 2 ) , as shown below. y ( 1 1 1 2 , 3 2 ) 30o 3 2 60o 1 1 2 x We will now repeat this process for a 45o reference angle. We first draw a right triangle that is based on a 45o reference angle, as shown below. y 1 ( x, y ) 1 y 45o x 1 x This triangle is a 45o-45o-90o triangle. We again want to find the values of x and y. Recall our theorem from the previous unit: In a 45o-45o-90o triangle, the legs are congruent, and the length of the hypotenuse is times the length of either leg. 2 Since the length of the hypotenuse is 2 times the length of either leg, we can say that the hypotenuse has length x 2 . But we know already that the hypotenuse has length 1, so x 2 = 1 . Solving for x, we find that x = 12 . Rationalizing the denominator, x= 1 2 ⋅ 2 2 = 2 2 . Since the legs are congruent, x = y , so y = Unit Circle Trigonometry 2 2 . Coordinates of Quadrantal Angles and First Quadrant Special Angles Based on the values of the sides of the triangle, we now know the coordinates of the point ( x, y ) where the terminal side of the 45o angle intersects the unit circle. This is the point ( 2 2 , 2 2 ) , as shown below. y 1 ( 1 2 2 , 2 2 ) 45o 2 2 45o x 1 2 2 Putting together all of the information from this section about quadrantal angles as well as special angles in the first quadrant, we obtain the following diagram: y ( 0, 1) 90o ( 1 2 , 60 1 3 2 ) o 45 ( −1, 0 ) ( 2 2 , 2 2 ) 30o ( 3 2 , 12 o 0o 180o 1 -1 360 ) (1, 0 ) o x -1 270o ( 0, − 1) We will use these coordinates in later sections to find trigonometric functions of special angles on the unit circle. Unit Circle Trigonometry Coordinates of Quadrantal Angles and First Quadrant Special Angles Definitions of the Six Trigonometric Functions We will soon learn how to apply the coordinates of the unit circle to find trigonometric functions, but we want to preface this discussion with a more general definition of the six trigonometric functions. Definitions of the Six Trigonometric Functions: General Case Let θ be an angle drawn in standard position, and let P ( x, y ) represent the point where the terminal side of the angle intersects the circle x 2 + y 2 = r 2 . The six trigonometric functions are defined as follows: sin (θ ) = y r csc (θ ) = 1 r = sin (θ ) y ( y ≠ 0) cos (θ ) = x r sec (θ ) = 1 r = cos (θ ) x ( x ≠ 0) tan (θ ) = y x cot (θ ) = 1 x = tan (θ ) y ( y ≠ 0) ( x ≠ 0) Alternative definitions for the tangent and the cotangent functions are as follows: tan (θ ) = sin (θ ) cos (θ ) cot (θ ) = cos (θ ) sin (θ ) The above functions are not really new to us if we relate them to our previous unit on right triangle trigonometry. For the purpose of remembering the formulas, we will choose to draw an angle θ in standard position in the first quadrant, and then draw a right triangle in the first quadrant which contains that angle, inscribed in the circle x 2 + y 2 = r 2 . (Remember that the circle x 2 + y 2 = r 2 is centered at the origin with radius r.) We label the horizontal side of the triangle x, the vertical side y, and the hypotenuse r (since it represents the radius of the circle.) A diagram of the triangle is shown below. Unit Circle Trigonometry Definitions of the Six Trigonometric Functions y x2 + y 2 = r 2 r y θ x x Notice that the formula x 2 + y 2 = r 2 (the equation of the circle) is simply the Pythagorean Theorem as it relates to the sides of the triangle. Recall the formulas for the basic trigonometric ratios which we learned in the previous unit on right triangle trigonometry, shown below in abbreviated form: Mnemonic: SOH-CAH-TOA sin(θ ) = Opposite Hypotenuse cos(θ ) = tan(θ ) = Adjacent Hypotenuse Opposite Adjacent and the reciprocal trigonometric ratios: csc(θ ) = 1 sin(θ ) = sec(θ ) = Hypotenuse Opposite 1 cos(θ ) = cot(θ ) = Hypotenuse Adjacent 1 tan(θ ) = Adjacent Opposite Using these formulas in the triangle from the diagram above, we obtain our six trigonometric functions which we apply to the coordinate plane: sin(θ ) = Opposite Hypotenuse = cos(θ ) = Adjacent Hypotenuse = tan(θ ) = Opposite Adjacent = y x Unit Circle Trigonometry y r csc(θ ) = 1 sin(θ ) = Hypotenuse Opposite = r y x r sec(θ ) = 1 cos(θ ) = Hypotenuse Adjacent = r x cot(θ ) = 1 tan(θ ) = Adjacent Opposite = x y Definitions of the Six Trigonometric Functions Note that even though we drew the right triangle in the first quadrant in order to easily relate these formulas to right triangle trigonometry, these definitions apply to any angle. (We will discuss later how to properly draw right triangles in other quadrants. Right triangles can not be drawn to illustrate the quadrantal angles, but the above formulas still apply.) Finally, let us justify the new formulas for tangent and cotangent. (We will do this algebraically, not with our right triangle picture from above.) Our alternative definition sin θ for the tangent ratio is tan (θ ) = cos((θ )) . If we substitute sin(θ ) = ry and cos(θ ) = rx into this equation, we can see that our alternative definition is equivalent to our definition tan(θ ) = xy . The algebraic justification is shown below: tan (θ ) = sin (θ ) cos (θ ) y y r y = r = ⋅ = x r x x r Similarly, our alternative definition for the cotangent ratio is cot (θ ) = cos (θ ) sin (θ ) . If we substitute sin(θ ) = ry and cos(θ ) = rx into this equation, we can see that our alternative definition for cotangent is equivalent to the formula cot(θ ) = tan(1θ ) = xy . The algebraic justification is shown below: cot (θ ) = cos (θ ) sin (θ ) x x r x = r = ⋅ = y r y y r Let us now apply our trigonometric definitions to the unit circle. Since the unit circle has radius 1, we can see that our trigonometric functions are greatly simplified: sin(θ ) = y y = =y r 1 csc (θ ) = r 1 = , y≠0 y y cos (θ ) = x x = =x r 1 sec (θ ) = r 1 = , x≠0 x x tan (θ ) = y sin (θ ) , x≠0 = x cos (θ ) cot (θ ) = x cos (θ ) , y≠0 = y sin (θ ) Unit Circle Trigonometry Definitions of the Six Trigonometric Functions