Year 10 Mathematics Core · Cambridge Ch 6, §6B · Mr Wong
Name
Class
Date
Learning intentions.
Use $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ to undo a trig ratio and find an angle
Choose the correct ratio from the two sides you've been given
Apply this to short worded problems by drawing the triangle first
Key results — inverse trig functions
If $\sin\theta=k$ then $\theta=\sin^{-1}(k)$ · If $\cos\theta=k$ then $\theta=\cos^{-1}(k)$ · If $\tan\theta=k$ then $\theta=\tan^{-1}(k)$
Check your calculator is in degree mode (DEG).
Warm-up — calculator
1. State the missing part of each sentence.
a If $\cos 60^\circ=0.5$, then $\cos^{-1}(0.5)=$ °.
b If $\sin 30^\circ=\tfrac{1}{2}$, then $\sin^{-1}\!\big(\;\,$ $\;\big)=30^\circ$.
c If $\tan 37^\circ\approx 0.75$, then $\tan^{-1}\!\big(\;\,$ $\;\big)\approx 37^\circ$.
2. Find $\theta$ to 2 d.p. where necessary.
a $\sin\theta=0.4$ $\theta=$
b $\cos\theta=0.5$ $\theta=$
c $\tan\theta=0.2$ $\theta=$
d $\sin\theta=0.1$ $\theta=$
Part A — Find the angle from a fraction of sides
Set up the inverse and write $\theta$ to 2 d.p.
3. Solve.
a $\sin\theta=\dfrac{3}{5}$ $\theta=$
b $\cos\theta=\dfrac{8}{17}$ $\theta=$
c $\tan\theta=\dfrac{7}{24}$ $\theta=$
d $\sin\theta=\dfrac{9}{15}$ $\theta=$
e $\cos\theta=\dfrac{2.4}{6}$ $\theta=$
f $\tan\theta=\dfrac{5}{12}$ $\theta=$
Trig 6B — continued
Part B (triangles) · Part C (applications)
Name
Part B — From the diagram
4. Find $\theta$ in each. Pick the ratio first from the two labelled sides, then apply the inverse. Round to 2 d.p.
(a) opp 1, hyp 2
(b) adj 1.5, hyp 2.5
(c) adj 5, opp 3
Part C — Applications
5. A long, straight mine tunnel is dug into the ground. Its final depth is $120$ m and the end of the tunnel is $100$ m horizontally from the entrance. Find the angle $\theta$ the tunnel makes with the horizontal, to 1 d.p.
6. A $4$ m ladder rests against a vertical wall, with its base $1.2$ m from the wall. Find the angle the ladder makes with the ground, to 1 d.p.
7. A skateboard ramp rises $0.8$ m over a horizontal run of $5$ m. Find the angle of the ramp, to 1 d.p.
8.Challenge. A kite is at the end of a $35$ m straight string. The kite is $28$ m vertically above the person holding the string. Find the angle the string makes with the ground, to 1 d.p. (Assume the string is straight.)