Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal side will intersect the unit circle at a particular point. The point of intersection will depend on the value of the angle. This implies that the x-coordinate and the y-coordinate of the point will depend on the value of the angle.,,, 5 50 0 5 0, 90 0 5, 0,,, 0 80 0 0, 0,, 7 5, 0 5 0 5 70 00 0, 5 0, Thus, for an angle that measures 0 in standard position, the terminal side will pass through the point, on the unit circle. For an angle that measures 70 in standard position, the terminal side will pass through the point 0, on the unit circle. Since the x- and y-coordinates of the point on the unit circle intersected by the terminal side depend on the measure of the angle, we can define each coordinate as a function of the angle. 7,,
Definition If θ is an angle in standard position and if the terminal side of θ intersects the unit circle at,, then sinθ sine of θ cosθ cosine of θ The sine and cosine functions belong to a class of functions that we call the trigonometric functions. Find the exact values of the sine and cosine functions for the given angles: Ex. a Ex. b 0 Ex. c Ex. d 0 a Since the terminal side of θ passes through, 0, then sinθ 0 and cosθ. b Since the terminal side of θ passes through sinθ and cosθ. c Since the terminal side of θ passes through then sinθ and cosθ.,, then,, d Since the terminal side of θ passes through, sinθ and cosθ., then There are four other trigonometric functions we can define in reference to our unit circle. We can first talk about the slope of the terminal side of θ in standard position. Two points on that line are 0, 0 and,, so the slope m 0 0. This value again depends on the value of θ, so we will define the tangent function tanθ to be equal to the slope of the terminal side. The other three trigonometric functions, the cosecant cscθ, the secant secθ, and the cotangent cotθ,are the reciprocals of the first three trigonometric functions sine, cosine, and tangent.
Definition If θ is an angle in standard position and if the terminal side of θ intersects the unit circle at,, then sinθ cosθ tanθ cscθ Note that cscθ secθ cotθ sinθ, secθ cosθ, and cotθ tanθ. Objective : Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle. Let P x, y be the point on the unit circle that corresponds to t. Find the values of the six trigonometric functions of t: Ex. a, 5 Ex. b 0, a sint tant csct sect cott 5 5 5 5 cost 5 5 5 5 5 5 5 b sint cost 0 tant csct sect 0 0 cott 0 0 which is undefined which is undefined 5 5 5
5 Objective : Find the Exact Value of the Six Trigonometric Functions of Quadrantal Angles. Find the exact values of the six trigonometric functions of: Ex. a θ 0 0 Ex. b θ 90 Ex. c θ 80 Ex. d θ 70 a The terminal side of θ intersects, 0 on the unit circle. Thus, sin0 0 cos0 tan0 0 0 csc0 0 which is undefined sec0 cot0 0 which is undefined b The terminal side of θ intersects 0, on the unit circle. Thus, sin cos 0 tan 0 which is undefined csc sec 0 which is undefined cot 0 0 c The terminal side of θ intersects, 0 on the unit circle. Thus, sin 0 cos tan csc cot 0 0 0 which is undefined sec 0 which is undefined d The terminal side of θ intersects 0, on the unit circle. Thus, sin cos 0 tan which is undefined csc 0 sec 0 which is undefined We can summarize our results: cot 0 0
θ θ sinθ cosθ tanθ cscθ secθ cotθ 0 0 0 0 undefined undefined 90 0 undefined undefined 0 80 0 0 undefined undefined 70 0 undefined undefined 0 Definition: Two angles are coterminal if they have the same initial and terminal sides. Properties of Coterminal Angles Two coterminal angles with differ by integer multiplies of 0 The Trig. value of two coterminal angles are equal. Find the following: Ex. a sin 50 Ex. b tan9 A coterminal angle to 50 is A coterminal angle to 9 50 + 0 70. is 9 So, sin 50 sin70 So, tan9 tan 0 Objectives & : Find the Exact Values of the Trigonometric Functions of for Special Angles The first special right triangle we want to examine is a 5-5 -90 right triangle. Since two of the angles are the same, that means two of the sides are equal, so we have an isosceles triangle. Also, the two equal sides of the isosceles triangle are the legs of the right triangle since the hypotenuse is always opposite of the right angle. The hypotenuse of the triangle will be equal to since we are on the unit circle. If we let a be the length of one of the equal sides, then we can the Pythagorean Theorem to find a. a + a a a take the square root a ignore answer 5 a 5 a
Thus, the terminal side of a 5 angle in standard position intersects the unit circle at,. The second special right triangle we want to examine is a 0-0 -90 right triangle. If we take an equilateral triangle and cut it in half along the altitude, we get two 0-0 -90 triangles. The hypotenuse of the triangle will be equal to since we are on the unit circle. The shortest side is half of the hypotenuse so it is /. If we let a be the length of the longer leg, then can use the Pythagorean Theorem to find the its length. 7 a + simplify a + a a Thus, we get two triangles: take the square root ignore the answer 0 0 a 0 0 Thus, the terminal side of a 0 angle in standard position intersects the unit circle at, and the terminal side of a 0 angle in standard position intersects the unit circle at,. Find the exact values of the six trigonometric functions of: Ex. 5a θ 0 Ex. 5b θ 5 0 0 Ex. 5c θ 0 a The terminal side of θ intersects, on the unit circle. Thus,
8 sin tan csc cot cos sec b The terminal side of θ intersects sin tan sec, on the unit circle. So, cos csc cot c The terminal side of θ intersects sin tan csc cot We can summarize our results:, on the unit circle. Thus, cos sec
9 θ θ sinθ cosθ tanθ cscθ secθ cotθ 0 5 0 Find the exact value of the following expression: Ex. a cos0 sin70 Ex. b cot cos Ex. c tan0 + csc0 Ex. d sin sin a cos0 sin70 b cot cos c tan0 + csc0 + 5 d sin sin Objective 5: Find the Exact Value of the Trigonometric Functions for Certain Integer Multiplies of the Special Angles. Since we know the values of the trigonometric functions for 0, 5, and 0, we can use symmetry to find the values of the trigonometric functions for 0, 0, 0, 5, 5, 5, 50, 0 and 00.
0, 5 50 0,, 7 0 0,, 5 5,, 5 5 5 7,
, 0 0,, 0 00 5, Find the exact values of the following: Ex. 7a cos5 Ex. 7b sin Ex. 7c tan Ex. 7d 7 csc Ex. 7e cot0 Ex. 7f sec50 a cos5 c tan d csc 7 e cot0 b sin f sec50
Objective : Approximate the Value of a Trigonometric Function. On a scientific calculator, SIN key is for the sine ratio, COS key is for the cosine ratio, and TAN key is for the tangent ratio. To evaluate a trigonometric ratio for a specific angle, you first want to be in the correct mode. If the angle is in degree, then the calculator needs to be in degree mode while if the angle is in radians, the calculator needs to be in radian mode. Next, type in the angle and then hit the appropriate trigonometric ratio. On some calculators, you might need to enter the trigonometric ratio first and then the angle. For the secant, cosecant, and cotangent, we want to use the following facts: secθ cosθ, cscθ sinθ, cotθ tanθ Evaluate the following round to three decimal places: Ex. 8a cos Ex. 8b tan78. Ex. 8c sin Ex. 8d csc5. Ex. 8e sec. Ex. 8f tan a We put our calculator in degree mode, type and hit the COS key: cos 0.8987 0.899 b We put our calculator in degree mode, type 78. and hit the TAN key: tan78..889.89 c We put our calculator in degree mode, type and hit the SIN key: sin 0.88 0.87 d We put our calculator in degree mode, type 5. and hit SIN key: sin 5. 0.808 Now, hit the /x key to take the reciprocal: csc5. sin5. 0.808....0.05 e Since there is no degree mark on the angle, then the angle is in radians. We put our calculator in radian mode, type. and hit the COS key: cos. 0. Now, hit the /x key to take the reciprocal: sec. cos. 0....7597.70
f Since there is no degree mark on the angle, then the angle is in radians. We put our calculator in radian mode, type and hit the TAN key: tan Error. This means that it is undefined. Objective 7: Use a Circle of Radius r to Evaluate the Trigonometric Functions. Now, consider an angle that is in standard position with a terminal side that intersects a circle of radius of r at the point x, y. By the Pythagorean Theorem, x + y r. Using similar triangles, it can be shown that: sinθ cscθ y r r y cosθ secθ x r r x tanθ cotθ y x Theorem Let θ be an angle in standard position whose terminal side intersects the circle x + y r at the point x, y. Then x y sinθ y r cosθ x r tanθ y x cscθ r y secθ r x cotθ x y Find the six trigonometric values of an angle θ with the following conditions: Ex. 9 5, is a point on the terminal side. First, we need to find the radius of the circle: r x + y r 5 + 5 + 9 r 9 ignore the answer Thus, x 5, y, and r : cosθ 5 5 sinθ cscθ secθ 5 5 tanθ 5 5 cotθ 5 5