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Important three-dimensional problem based on pyramid IGCSE/GCSE Trigonometry 0580 Extended

Question: The diagram shows a pyramid with base ABC. CD is perpendicular to both CA and CB. ∟CBD=34°, ∟ADB=45°, ∟DBA=60°, BC= 20 cm. Calculate the size of the angle between the line AD and the plane ABC. Write the answer correctly to 1 dp.

Solution: Finding the size of the angle between the line AD and the plane ABC means 

finding ∟θ. 

So, the question here is to find the size of ∟θ. 

For finding ∟θ, we require DC and AD sides of ∆ADC.

We can find DC and DB sides from the right ∆DCB,

DC/20=tan 34° DC=20× tan 34°=13.5 cm 

 20/DB=cos 34° 

 DB=20/cos 34° =24.1 cm 

For finding side AD we will be using the sine rule in ∆ADB.

∟DAB=180°-60°-45°=75° 

 b/sin 60° = 24.1/sin 75° 

 b=AD = (24.1×sin 60°)/sin 75° =21.6 cm 

 So, in ∆DAC, we have AD=21.6 cm, DC=13.5 cm 

 Sin θ=DC/AD=13.5/21.6=0.625 

 ∟θ=sin⁻¹ 0.625 =38.7°

Answer 38.7°

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