Download Compound Angles and Double Angles File

Trigonometry II: Compound Angles and Double Angles
2.
Do not use the buttons for trigonometric functions or inverse trigonometric functions on your
calculator to solve the following
3
(i) If sin 𝐴 = 5 and if 𝐴 is an acute angle, use Rules 𝐼(a) to discover sin (𝐴 +
𝐴)= sin 2A 𝑎𝑛𝑑 cos (𝐴 + 𝐴) = cos 2A
4
24
(ii) If sin 𝐴 = 5 and sin 𝐵 = 25 and if 𝐴 and 𝐵 are acute angles, discover the value of
tan(𝐴 + 𝐵)
(ii) If tan(𝑥 + 45𝑜 )tan𝑥 = 3, discover the possible values for tan 𝑥.
(iv) Simplify and thereby solve
(v) If tan 𝑥 =
√3+1
1− √3
√3
2
1
cos 750 + 2 sin 75𝑜
discover the value of 𝑥 given that
90𝑜 < 𝑥 < 180°
3.
Simplify sin (𝐴 + 𝐵) + sin (𝐴 − 𝐵). In doing so, discover all the values of 𝑥 between 0𝑜 and
3600 which satisfy sin (𝑥 + 60𝑜 ) + sin (𝑥 − 60𝑜 ) =
4.
1
√2
Write the expansion for tan (𝛩 + 𝛩). In doing so, discover an expression for tan 2𝛩 in terms of
tan 𝛩.
5.
Solve the equations for the values of 𝑥 between 0𝑜 and 360𝑜 , using your calculator when
necessary
(i)
2 cos 𝑥 = sin (𝑥 + 60𝑜 )
(ii) sin (𝑥 + 45𝑜 ) = sin 𝑥
(iii) sin(𝑥 + 30°) =
1
cos 𝑥
2
(iv) 4 cos (𝑥 + 10𝑜 ) = 3 sin (𝑥 − 10𝑜 )
4.2
Double angle formulae
For your convenience, Rules I – III given in section 4.1 are listed below.
sin (𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
,(a)
cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
,(b)
sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − 𝑐𝑜𝑠𝐴 sin 𝐵
,(c)
cos(𝐴 − 𝐵) − cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
tan(𝐴 + 𝐵) =
tan 𝐴+tan 𝐵
1−tan 𝐴 tan 𝐵
tan(𝐴 − 𝐵) =
tan 𝐴−tan 𝐵
1+tan 𝐴 tan 𝐵
,(d)
,(e)
,(f)
}
Rules I
}
Rules II
}
Rules III