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In die diagram is P(3 ; t) 'n punt in die Cartesianse vlak - NSC Mathematics - Question 5 - 2017 - Paper 2

Question 5

In die diagram is P(3 ; t) 'n punt in die Cartesianse vlak. OP = √34 en HΩP - β is 'n inspirgende (refleks-) hoek. Sonder die gebruik van 'n sakrekenaar, bepaal die... show full transcript

Worked Solution & Example Answer:In die diagram is P(3 ; t) 'n punt in die Cartesianse vlak - NSC Mathematics - Question 5 - 2017 - Paper 2

Step 1

5.2.1 t

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Answer

As OP = ( \sqrt{34} ), dan kan ons die waarde van t bereken deur die pitagoras te gebruik:

[ t = \sqrt{OP^2 - 3^2} = \sqrt{34 - 9} = \sqrt{25} = 5 ]

Step 2

5.2.2 tan β

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Answer

Die tangens van β kan soos volg bereken word:

[ tan \beta = \frac{tegnis}{aanliggend} = \frac{t}{3} = \frac{5}{3} ]

Step 3

5.2.3 cos 2β

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Answer

Die kosinus van 2β gebruik die formule:

[ cos 2\beta = 2 \cos^2 \beta - 1 ]

Eerstens moet ons (\cos \beta) bepaal:

[ \cos \beta = \frac{3}{\sqrt{(3^2 + 5^2)}} = \frac{3}{\sqrt{34}} ]

Dan kan ons dit in die oorspronklike formule invul:

[ cos 2\beta = 2 \left(\frac{3}{\sqrt{34}}\right)^2 - 1 = 2 \cdot \frac{9}{34} - 1 = \frac{18}{34} - 1 = -\frac{16}{34} = -\frac{8}{17} ]

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