2An angle has “a positive” sign if its rotation is anticlockwise Terminal sideInitial sideAngle is defined a rotation result from initial side to the terminal sideAn angle has “a positive” sign if its rotation is anticlockwiseAn angle has “a negative” sign if its rotation is clockwiseWe only talk about the magnitude of angle~ not observe its signsTerminal side
3What is a radian? radius arc=radius o Angle = 1 rad radius Why do mathematicians use radians instead of degrees?
4How many times does the radius divide into the circumference? There are 2 radians in a circle.1 radian = = 57.3o
9Objectives/Assignment Find the since, the cosine, and the tangent of an acute triangle.Use trionometric ratios to solve real-life problems
10Finding Trig RatiosA trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sin, cos, and tan
11Trigonometric RatiosLet ∆ABC be a right triangle. The since, the cosine, and the tangent of the acute angle A are defined as follows.Side adjacent to Abcos A ==hypotenusecSide opposite Aasin A ==hypotenusecSide opposite Aatan A ==Side adjacent to Ab
12Exercise:In the PQR triangle is right angled at R happens cos P = 8/ 17…find the value of tg P and tg Q ?
13Exercises1. PQR triangle is right angled at R & sin P cosQ = ⅗. Find out the valuecABC triangle is right angle at A. BC=p, AD is perpendicular at BC, DE perpendicular at AC, angle B = Q, prove that : DE=psin2 Qcos Q2DEABC triangle is rightangle at A. BC=p, AD is perpendicular at BC, DE perpendicular at AC, angle B = Q, prove that : DE=psin2 Qcos QAB
14Evaluating Trigonometric Functions Acute angle A is drawn instandard position as shown.Right-Triangle-Based Definitions of Trigonometric FunctionsFor any acute angle A in standard position,
15Finding Trigonometric Function Values Finding Trigonometric Function Values of an Acute Angle in a Right TriangleExample Find the values of sin A, cos A, and tan Ain the right triangle.Solutionlength of side opposite angle A is 7length of side adjacent angle A is 24length of hypotenuse is 25
16Trigonometric Function Values of Special Angles Angles that deserve special study are 30º, 45º, and 60º.Using the figures above, we have the exact values of the special angles summarized in the table on the right.
17Let’s try to prove it ! OA=OB OA2 + OB2 = OC2 OA2 + OA2 = r2 2OA2 = 1 X45OOA=OBOA2 + OB2 = OC2OA2 + OA2 = r22OA2 = 1OA2 = ½= OBHow about 300 , 600 and 900OA =How about 300 , 600 and 900
18Cofunction Identities In a right triangle ABC, with right angle C, the acute angles A and B are complementary.Since angles A and B are complementary, andsin A = cos B, the functions sine and cosine are called cofunctions. Similarly for secant and cosecant, and tangent and cotangent.
19Cofunction Identities If A is an acute angle measured in degrees, thenIf A is an acute angle measured in radians, thenNote These identities actually apply to all angles (not just acute angles).
20Reference AnglesA reference angle for an angle , written , is the positive acute angle made by the terminal side of angle and the x-axis.Example Find the reference angle for each angle.218º (b)Solution(a) = 218º – 180º = 38º (b)
21Special Angles as Reference Angles Example Find the values of the trigonometric functions for 210º. Solution The reference angle for 210º is 210º – 180º = 30º.Choose point P on the terminal side so that the distance from the origin to P is 2. A 30º - 60º right triangle is formed.
22Finding Trigonometric Function Values Using Reference Angles Example Find the exact value of each expression.cos(–240º) (b) tan 675ºSolution–240º is coterminal with 120º.The reference angle is180º – 120º = 60º. Since –240ºlies in quadrant II, thecos(–240º) is negative.Similarly,tan 675º = tan 315º= –tan 45º = –1.
23Finding Angle MeasureExample Find all values of , if is in the interval[0º, 360º) andSolution Since cosine is negative, must lie ineither quadrant II or III. SinceSo the reference angle = 45º.The quadrant II angle = 180º – 45º = 135º, and thequadrant III angle = 180º + 45º = 225º.
24Ex. 1: Finding Trig Ratios Compare the sine, the cosine, and the tangent ratios for A in each triangle beside.By the Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion which implies that the trigonometric ratios for A in each triangle are the same.
25Ex. 1: Finding Trig Ratios LargeSmallopposite8sin A =≈0.47064≈0.4706hypotenuse178.5adjacent7.5cosA =15≈0.8824≈0.8824hypotenuse8.517oppositetanA =84≈0.5333≈0.5333adjacent157.5Trig ratios are often expressed as decimal approximations.
26Ex. 2: Finding Trig Ratios opposite5sin S =≈0.3846hypotenuse13adjacentcosS =12≈0.9231hypotenuse13oppositetanS =5≈0.4167adjacent12
27Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle. opposite12sin S =≈0.9231hypotenuse13adjacentcosS =5≈0.3846hypotenuse13oppositetanS =12≈2.4adjacent5
28Notes:If you look back, you will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.
29Using Trigonometric Ratios in Real-life Suppose you stand and look up at a point in the distance. Maybe you are looking up at the top of a tree as in Example 6. The angle that your line of sight makes with a line drawn horizontally is called angle of elevation.
30Ex. 6: Indirect Measurement You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.
31The math The tree is about 76 feet tall. Write the ratio tan 59° =oppositeadjacentWrite the ratiotan 59° =h45Substitute valuesMultiply each side by 4545 tan 59° = hUse a calculator or table to find tan 59°45 (1.6643) ≈ hSimplify75.9 ≈ hThe tree is about 76 feet tall.
32Ex. 7: Estimating Distance Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet.30°
33Now the mathsin 30° =oppositehypotenuse30°Write the ratio for sine of 30°sin 30° =76dSubstitute values.d sin 30° = 76Multiply each side by d.sin 30°76d =Divide each side by sin 30°0.576d =Substitute 0.5 for sin 30°d = 152SimplifyA person travels 152 feet on the escalator stairs.