In the diagram, B is 12 m due east of A.
C is 13 m from A and due south of B. Use the following information to answer the questions:
${\sin }^{-1}\left(\frac{12}{13}\right)={67.4}^o$, ${\cos }^{-1}\left(\frac{12}{13}\right)={22.6}^o$, ${\tan }^{-1}\frac{12}{13}={42.7}^o$
(a) Find the length, x.

(b) Calculate angle ACB.

(c) Find the bearing of C from A.

The diagram shows a right angled triangle ABC.
AC = 12 cm and angle BAC = 60°.
[sin 60 = 0.866, cos 60 = 0.5, tan 60 = 1.732] Calculate:
(a) The area of the triangle,

(b) The shortest distance from B to AC.

Points C and G lie on horizontal line HE as illustrated in the diagram.
Lines BH and DG are vertical.
BC = 80 cm, HC = 60 cm, GE = 35 cm, DG = 40 cm, and angle DCG = 32°.
Use any of the information in the table where necessary.

sin−1 (34) = 48.6°	tan32° = 0.623	cos−1 (78) = 29.0°
cos−1 (34) = 41.4°	sin32° = 0.530	tan−1 (78) = 41.2°
tan−1 (34) = 36.9°	cos32° = 0.848	sin−1 (78) = 61.0°

(a) Calculate:
(i) The size of the angle HBC,

(ii) The length of line CD, leaving the answer as a fraction.

(iii) Write the angle of depression of C from D.

Work out the length AC.

The diagram shows the right-angled triangle ABC.
D is a point on BC produced.
AB = 5 cm and BC = 12 cm. (a) Work out the length AC.

(b) Write down the value of the cosine of angle ACD.

The diagram shows a right-angled triangle ABC.
AB = 15 cm and BC = 8 cm. Calculate:
(a) the length AC,

(b) the area of triangle ABC.

In the diagram, A, B, C, and D are points on a horizontal ground.
AB = 6 m, AC = 5.6 m and CD = 4.3 m.
Angle BAC = 35° and angle ADC = 90°. (a) What is the name of the polygon ABCD?

(b) Calculate:
(i) AD,

(ii) BC.

(c) ABPQ is a vertical wall of height 2.2 m.
The wall is built on the side AB of the polygon ABCD.
A bird walks from Q to P.
Calculate the greatest angle of elevation of the bird from C.

The diagram shows triangle PRS.
PS = 15 cm, QS = 12 cm, RS = 17 cm and $Q\hat{S}R=53°$. Calculate:
(a) QR,

(b) The angle PSQ,

(c) The shortest distance from Q to PS.

The diagram represents a field, ABCD in the shape of a quadrilateral.
AC is a straight footpath across the field and DC is parallel to AB.
AB = 70 m, BC = 79 m and $A\hat{B}C=120°$. (a) Calculate the distance AC.

(b) Calculate the area of triangle ABC.

(c) A farmer walks the shortest distance from point B to line CD.
Calculate this distance.

The diagram shows a quadrilateral ABCD.
AB = 4 cm, CD = 11 cm, $B\hat{C}D=25°$ and $B\hat{A}D=\theta$. (a) Show that $BD=\sqrt{16(5-4\cos \theta )}$ cm.

(b) Let θ = 40°.
(i) Find the value of $\sin CBD$.

(ii) Find the value of the angle CBD.

(iii) Work out the perimeter of ABCD.

(iv) Find the area of triangle ABD.