Solutions for Trigonometric Functions of Any Angle

Transcription

1 Solutions for Trigonometric Functions of Any Angle I. Souldatos Answers Problem... Consider the following triangle with AB = and AC =.. Find the hypotenuse.. Find all trigonometric numbers of angle B.. Find all trigonometric numbers of angle C.. Decide which trigonometric numbers of B are equal to a trigonometric number of C. Solution () Use the Pythagorean Theorem: BC = + =. (,) Use the definitions: sin(b) = cos(b) = tan(b) = cot(b) = sec(b) = csc(b) =

2 ANSWERS and sin(c) = cos(c) = tan(c) = cot(c) = () The following are equal: sec(c) = csc(c) = sin(b) = cos(c) cos(b) = sin(c) tan(b) = cot(c) cot(b) = tan(c) sec(b) = csc(c) csc(b) = sec(c) The reason is the B, C are complementary, i.e. C = 9 o B or B + C = 9 o. Complementary angles result in the complementary functions (sine-cosine, tangent-cotangent, secantcosecant) being equal,. Grade yourself: One point for part (), one point for each trigonometric number in parts () and (), and one point for every formula in part (). Subtotal: /9... Consider the following triangle with AB = and B = o. Solve the triangle, i.e. find all sides and all angles. Page of

3 ANSWERS Solution C = 9 o o = o. We can find AC using tan(b). tan(b) = AC AB tan( o ) = AC AC = tan( o ).89 =.9 There are many ways to find BC, e.g. Pythagorean Theorem or use a trigonometric number of B or C. Let s use cos(b). cos(b) = AB BC cos( o ) = BC BC cos( o ) = BC = cos( o ).7 =.7 Grade yourself: One point for each of C, AC, BC. Subtotal: /... Assume that θ is in the II quadrant and tan(θ) =. Find all the trigonometric numbers of θ. Then find all the trigonometric numbers of θ and all the trigonometric numbers of 8 θ. Solution For every trigonometric number of θ we use the appropriate equation. Cotangent: cot(θ) = tan(θ) =. Secant: tan (θ) + = sec (θ) ( ) + = sec (θ) = sec (θ) sec(θ) = ± = ± In the II quadrant, is negative. Therefore, sec(θ) is also negative and sec(θ) =. Cosine: = sec(θ) =. Page of

4 ANSWERS Sine: Cosecant: tan(θ) = sin(θ) tan(θ) = sin(θ) sin(θ) = ( ) = csc(θ) = sin(θ) =. For θ and 8 o θ we use the corresponding formulas: sin( θ) = sin(θ) = cos( θ) = + = tan( θ) = tan(θ) = + cot( θ) = cot(θ) = + sec( θ) = + sec(θ) = and csc( θ) = csc(θ) = sin(8 o θ) = + sin(θ) = + cos(8 o θ) = = + tan(8 o θ) = tan(θ) = + cot(8 o θ) = cot(θ) = + sec(8 o θ) = sec(θ) = + csc(8 o θ) = + csc(θ) = + Grade yourself: One point for each trigonometric number of θ, θ and 8 o θ. Subtotal: /7... (a) Fill in the table by memory, not using a calculator. Page of

5 ANSWERS θ o o o o 9 o 8 o 7 o sin(θ) = = = = = = tan(θ) = sin(θ) = = = = : undefined = Grade yourself: One point for each box. Subtotal: /. : undefined (b) Convert degrees to radians, and radians to degrees. o = π 8 = π radians radians= 8 π = π. degrees Grade yourself: One point for each answer. Subtotal: /. (c) Find the reference angle for o. This angle is in the II quadrant. The reference angle is 8 o o = 7 o. o. This angle is in the III quadrant. The reference angle is o 8 o = o. o. This angle is in the IV quadrant. The reference angle is o o = o. radians. This angle is in the IV quadrant. The reference angle is π.8 radians. Grade yourself: One point for each answer. Subtotal: /. (d) Find the angle between and that is coterminal with angle. We substract o from o enough times, so that the resulting angle is between and. In this problem it suffices to substract twice. So, o is coterminal with 8 o. o o = o 7 o = 8 o. Grade yourself: One point for the right answer. Subtotal: /. (e) Find the angle between and π radians that is coterminal with π radians angle. Again, we substract π from π enough times until the resulting angle is between and π radians. Here is suffices to substract once. So, π is coterminal with π. π π = π. Grade yourself: One point for the right answer. Subtotal: /. Subtotal for.: /9 Page of