Unit 4 Section 9 - 4.9 SOLUTION OF RIGHT TRIANGLES...

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4.9 SOLUTION OF RIGHT TRIANGLES Trigonometric Functions Definition 4.9.1 Let be an angle (not quadrantal) whose vertex is at the origin and the initial side is the positive side of the -axis (standard position) and let be any point on the terminal side of the angle distinct from the origin. Also, let be the distance of point from the origin (see Figure 1). The six trigonometric functions (sine, cosine, tangent, cotangent, secant and cosecant) are defined as follows: Figure 1 Trigonometric Functions of Acute Angles In dealing with any right triangle, it will be convenient to denote the vertices as and with the vertex of the right angle; and to denote the sides opposite the angles; as and , respectively. With respect to angle , will be called the opposite side and will be called the adjacent side ; with respect to angle , will be called the opposite side and the adjacent side . Side will always be called the hypotenuse .
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Figure 2 Definition 4.9.2 If now the right triangle is placed in a coordinate system so that angle is in standard position (Figure 2), the point on the terminal side of angle has coordinates and the distance then the trigonometric functions of angle may be defined in terms of the sides of the right triangle, as follows: Trigonometric Functions of Complementary Angles Definition 4.9.3 The acute angles and of the right triangle are complementary; that is, . From Fig. 1, we have Example 4.9.1 Find the values of the trigonometric functions of the angles of the right triangle in Fig. 3. Figure 3 Note that the sine of an angle is also the cosine of the angle’s complement. Similarly, the tangent of an angle is the cotangent of the angle’s complement, and the secant of an angle is the cosecant of
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the angle’s complement. These pairs of functions are called cofunctions . A list of cofunction identities follows. Cofunction Identities Let be an acute interior angle of a right triangle. Then Trigonometric Functions of 30 o , 45 o , and 60 o In circular functions, we call the arc lengths or real numbers whose circular function values can be computed exactly as special real numbers. In trigonometric functions, we call them special angles . 30 o , 45 o , and 60 o are the special acute angles. Their trigonometric function values are equal to the circular function values of their radian counterparts. Trigonometric Function Values In general, the trigonometric function values of angle measures (in degrees) are equal to their arc length counterparts (in radians). Solving Right Triangles
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This note was uploaded on 05/13/2011 for the course MATH 17 taught by Professor Dikopaalam during the Spring '11 term at University of the Philippines Los Baños.

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Unit 4 Section 9 - 4.9 SOLUTION OF RIGHT TRIANGLES...

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