Trigonometric functions and circular measure - the Australian ...

More documents

Recommendations

Info

{46} • Trigonometric functions and circular measureAnswers to exercisesExercise 1Coordinates of P θ sinθ cosθ tanθ(1,0) 0 ◦ 0 1 0(0,1) 90 ◦ 1 0 undefined(−1,0) 180 ◦ 0 −1 0(0,−1) 270 ◦ −1 0 undefined(1,0) 360 ◦ 0 1 0Exercise 2a sin330 ◦ = −sin30 ◦ = − 1 23cos330 ◦ = cos30 ◦ =2tan330 ◦ = −tan30 ◦ = − 1 3b If θ is in the fourth quadrant, the related angle is 360 ◦ − θ.Exercise 3a sin210 ◦ = − 1 2b cos315 ◦ = 1 2c tan150 ◦ = − 1 3Exercise 4a In △BC P, we have h = a sinB. In △AC P, we have h = b sin A.Hence a sinB = b sin A =⇒asin A = bsinB .b In △BC M, we have h = a sinB. In △AC M, we have h = b sin(180 ◦ − A) = b sin A.aAgain it follows thatsin A = bsinB .Exercise 5Using A = 1 2ab sinθ, we have5 = 1 2 × 5 × 4 × sinθ =⇒ sinθ = 1 2=⇒ θ = 30 ◦ ,150 ◦ .
A guide for teachers – Years 11 and 12 • {47}5 cm5 cm30 º150 º4 cm 4 cmExercise 612 ac sinB = 1 2Exercise 7LHS =Exercise 8bc sin A =⇒ a sinB = b sin A =⇒asin A =1secθ + tanθ = 11cosθ + sinθcosθ= cosθ1 + sinθ = RHSbsinBThe double angle formula for cosine gives cos30 ◦ = 1 − 2sin 2 15 ◦ . Sosin 2 15 ◦ = 1 2= 1 2Hence sin15 ◦ = 1 2(1 − cos30◦ )(1 −3)2= 1 4(2 −3).√2 − 3. (Take the positive square root, as sin15 ◦ is positive.)An alternative method is to start from sin30 ◦ = 2sin15 ◦ cos15 ◦ . Proceed by squaring bothsides of the equation and using the Pythagorean identity.Exercise 9Using the double angle formulas, we havetan2θ = sin2θcos2θ = 2sinθ cosθcos 2 θ − sin 2 θ .If we assume that tanθ is defined, then we can divide the numerator and denominatorby cos 2 θ to obtaintan2θ =2 sinθcosθ1 − sin2 θcos 2 θ= 2tanθ1 − tan 2 θ .Note that tan2θ is defined if and only if cos2θ = cos 2 θ − sin 2 θ ≠ 0, in which case tanθcannot equal ±1.
Page 1 and 2: 1Supporting Australian Mathematics
Page 3 and 4: Assumed knowledge . . . . . . . . .
Page 5 and 6: A guide for teachers - Years 11 and
Page 45: A guide for teachers - Years 11 and
Page 51: 0 1 2 3 4 5 6 7 8 9 10 11 12

Trigonometric functions and circular measure - the Australian ...

Create successful ePaper yourself

Delete template?

Save as template?