Gcse H3 02g4 02 3d Trigonometry 53164j

For use only in Dr Challoner's Grammar School GCSE-H3-02g4-02 3D Trigonometry

3D Trigonometry a 2 = b 2 + c 2 − 2bc cos A and cos A =

1.

b2 + c2 − a 2 2bc

ABCDEFGH is a cuboid with AB = 25cm, AD =13cm and AE =18cm. (a) Which of the following angles are right angles ABC ABG AFH ADG HFC DGF? (b) Find the following lengths (leave as square roots) (i) AC (ii) EB (iii) BG (iv) DF

G E 18cm

F D 13cm

A

C

25cm

B

Show that the angle GAC is 32.6o (to 1dp). Calculate the angle BEC (to 1dp) using trigonometry. Calculate EBG (to 1dp) using the Cosine rule.

(c) (d) (e)

2. E

C

D 20cm

24cm M

A

35cm

B

In the pyramid ABCDE, the rectangular base ABCD is horizontal and EM is vertical. M is the midpoint of AC. AB = 35cm, AD = 20cm and AE = 24cm. Find the following lengths (to 3sf) (a) AM (i) EM (ii) Find the following angles (to 1dp) (b) EAM (i) AEB (ii) (iii) DEB (i.e. 2 × ∠DEM )

PTO

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For use only in Dr Challoner's Grammar School GCSE-H3-02g4-02 3D Trigonometry 3.

The diagram below represents a gift box. ABCD and EFGH are horizontal squares of side lengths 12cm and 20cm respectively. The 4 slanting sides AE, BF, CG and DH all are 28cm long and all make the same angle with the horizontal base. H 20cm G

E

F 28cm D C

A

B

12cm

Let X be the point on EF such that AX is perpendicular to EF. X F E

A (a) (b) (c) (d)

B

Explain why the length of EX is 4cm. Calculate the angle AEF (to 1dp). If Y is the point on the top of the box which is vertically above B then find FY (to 3sf). (HINT: Consider a bird’s eye view of EFGH) Hence find the vertical height of the gift box (to 3sf). (HINT: Draw BYF)

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