Presentation on theme: "10 Trigonometry (1) Contents 10.1 Basic Terminology of Trigonometry"— Presentation transcript:
110 Trigonometry (1) Contents 10.1 Basic Terminology of Trigonometry 10.2 Definition of Trigonometric Ratio of Any Angles10.3 Finding Trigonometric Ratios Without Using a Calculator10.4 Trigonometric Equations10.5 Graphs of Trigonometric FunctionsHome10.6 Graphical Solutions of Trigonometric Equations
210.1 Basic Terminology of Trigonometry A. Angles of RotationFig 10.3 shows a circle on a rectangular coordinate plane, with centre at the origin O and of radius r. A is a point on the circle.Fig. 10.3Fig. 10.3An angle θ is formed by rotating OA about the origin O to reach OP as shown in Fig The angle θ is called the angle of rotation.We call OA the initial side of the angle and OP the terminal side of the angle.
310.1 Basic Terminology of Trigonometry B. QuadrantsIn a rectangular coordinate plane, the x-axis and y-axis divide the plane into four regions which are known as quadrants. They are labelled and are shown in FigFig. 10.7The x-axis and y-axis do not belong to any of the four quadrants.
410.1 Basic Terminology of Trigonometry An angle of rotation θ is said to lie in a quadrant if its terminal side lies in that quadrant.When the measure of θ is either negative or greater than 360º, the position of the terminal side of θ determines the quadrant in which it lies.For example, the terminal side of –240º coincides with that of 120º as shown in FigFig. 10.8Since 120º lies in quadrant II, –240º also lies in quadrant II.
510.2 Definition of Trigonometric Ratio of Any Angles A. DefinitionThe trigonometric ratios of an acute angle are defined as the ratios between the sides of a right-angled triangle.To define the trigonometric ratios for an arbitrary angle, we can first place triangle OPQ on a rectangular coordinate plane as shown in FigThe trigonometric ratios of the angle then can be defined in terms of x, y and r, whereFig
610.2 Definition of Trigonometric Ratio of Any Angles For angles greater than 90o, we can consider the terminal side OP of the angle θ and let the coordinates of P be (x, y):θ lies in quadrant IIθ lies in quadrant IIIθ lies in quadrant IVFig (a)Fig (b)Fig (c)Then the trigonometric ratios of θ are defined in terms of x, y and r,as follows:where
710.2 Definition of Trigonometric Ratio of Any Angles The following identities hold for all measures of θ.(a)(b)Proof of (a):By Pythagoras’ Theorem,Proof of (b):
810.2 Definition of Trigonometric Ratio of Any Angles B. Signs of the Three Trigonometric RatiosThe signs of the three trigonometric functions can be summarized in Fig This figure is also called a ‘ASTC’ diagram.Notes:In the ‘ASTC’ diagram, the letters A, S, T and C indicate the trigonometric ratios that are positive, whereA: All positives;S: Sin positives;T: Tan positives; andC: Cos positives.Fig
910.3 Finding Trigonometric Ratios Without Using a Calculator A. Trigonometric Ratios of Angles Formed by the Coordinate AxesFig (a)Fig.10.27(c)Fig.10.27(b)Fig.10.27(d)
1010.3 Finding Trigonometric Ratios Without Using a Calculator The values of the trigonometric ratios of,andcan be found and the results are listed in Table 10.3.θsin θcos θtan θ0º190ºundefined180º–1270º360ºAs shown in Fig (d), the terminal sides OP of and lie in the same position. Hence the trigonometric ratios of and are the same.Table 10.3
1110.3 Finding Trigonometric Ratios Without Using a Calculator B. Trigonometric Ratios by Considering the Reference AnglesThe positive acute angle formed between the terminal side of the angle and the x-axis is called reference angle as shown in Fig (a) – (d). lies in quadrant I lies in quadrant III lies in quadrant II lies in quadrant IVFig (a)Fig (c)Fig (b)Fig (d)
1210.3 Finding Trigonometric Ratios Without Using a Calculator C. Finding Trigonometric Ratios when other Trigonometric Ratios are GivenExample 10.3TIf, where,findand.Solution:Sinceand θ lies in quadrant II, we takeSince θ lies in quadrant II, then P(–2, 3) is a point on the terminal side of θ as shown in the figure.
1310.3 Finding Trigonometric Ratios Without Using a Calculator Consider (–2, 3) to be a point on the terminal side of θ. Then use the definitions to find other trigonometric ratios of .By definition,
1410.4 Trigonometric Equations A. Inverses of the Trigonometric RatiosIf a trigonometric ratio of an angle is given, then we can find the angle. This process is called finding the inverse of the trigonometric ratio.
1510.4 Trigonometric Equations Example 10.5TIf, where,find θ.Solution:Since tan θ < 0, θ must lie in either quadrant II or quadrant IV.Step 1:Step 2:Let β be the reference angle of θ. ThenStep 3:By locating θ and its reference angle diagrammatically as shown in the figures,or
1610.4 Trigonometric Equations B. Algebraic Solutions of Trigonometric EquationsAn equation involving trigonometric ratios of an unknown angle is called a trigonometric equation.For example, (2 sin x – 1)(sin x + 1) = 0 is a trigonometric equation.It is not an identity because it holds for some values of x only, such as x = 30º. x = 30º is called a solution of this trigonometric equation.
1710.4 Trigonometric Equations Example 10.7TSolve the equationfor(Give the answers correct to 1 decimal place.)Solution:If cos = 0, thensin = 1 , which does not satisfy 2sin + cos = 0.As cos 0, we can divide the expression by cos .(correct to 1 decimal place)
1810.5 Graphs of Trigonometric Functions A. The Graph of y = sin xx0o30o60o90o120o150o180o210o240o270o300o330o360oy0.50.91–0.5–0.9–1Table 10.4The ordered pairs in Table 10.4 are plotted and the graph of y = sin x is drawn and shown in FigFig
1910.5 Graphs of Trigonometric Functions Properties of Sine Function:FigMaximum value of sin x = 1 , which corresponds to x = 90o .Minimum value of sin x = –1 , which corresponds to x = 270o .The curve of y = sin x forrepeats itself in every 360o.
2010.5 Graphs of Trigonometric Functions B. The Graph of y = cos xx0o30o60o90o120o150o180o210o240o270o300o330o360oy10.90.5–0.5–0.9–1Table 10.5Similarly, cos x repeats itself in every 360o. cos x is a periodic function with a period of 360o andfor all values of x.Fig
2110.5 Graphs of Trigonometric Functions C. The Graph of y = tan xx0o30o60o75o90o105o120o150o180oy0.61.73.7undefined–3.7–1.7–0.6x210o240o255o270o285o300o330o360oy0.61.73.7undefined–3.7–1.7–0.6Table 10.6Fig
2210.5 Graphs of Trigonometric Functions Properties of tangent function :Fig, the tangent function exhibits the following behaviour:1. ForFrom 0o to 90o , tan x increases from 0 to positive infinity.From 90o to 180o , tan x increases from negative infinity to 0.2. tan x is a periodic function with a period of 180o.
2310.6 Graphical Solutions of Trigonometric Equations To solve the equation from the graph, draw a straight line as shown below.The straight line cuts the graph of at P and Q.The approximate solution is 228 or 312.To solve the equation from the graph, draw a straight line as shown below.The straight line does not intersect with the graph of