NSC Mathematics Trigonometry: 6.1 Vereenvoudig, sonder die ge

6.1 Vereenvoudig, sonder die gebruik van 'n sakrekenaar, die volgende uitdrukking tot 'n enkele trigonometriese term: \[ \frac{\sin 10^{\circ} + \tan(360^{\circ} - \theta) \cdot \sin 20^{\circ}}{\cos 440^{\circ}} \] 6.2 Gegee: $ \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = k \cos 2x $ 6.2.1 Bereken die waarde van $ k $ as $ \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = k \cos 2x $ - NSC Mathematics - Question 6 - 2022 - Paper 2

Question 6

$6.1--Vereenvoudig,-sonder-die-gebruik-van-'n-sakrekenaar,-die-volgende-uitdrukking-tot-'n-enkele-trigonometriese-term:--\[-\frac{\sin-10^{\circ}-+-\tan(360^{\circ}---\theta)-\cdot-\sin-20^{\circ}}{\cos-440^{\circ}}-\]--6.2--Gegee:-$-\sin(60^{\circ}-+-2x)-+-\sin(60^{\circ}---2x)-=-k-\cos-2x-$--6.2.1--Bereken-die-waarde-van-$-k-$-as-$-\sin(60^{\circ}-+-2x)-+-\sin(60^{\circ}---2x)-=-k-\cos-2x-$-NSC Mathematics-Question 6-2022-Paper 2.png$

6.1 Vereenvoudig, sonder die gebruik van 'n sakrekenaar, die volgende uitdrukking tot 'n enkele trigonometriese term: \[ \frac{\sin 10^{\circ} + \tan(360^{\circ} -... show full transcript

Worked Solution & Example Answer:6.1 Vereenvoudig, sonder die gebruik van 'n sakrekenaar, die volgende uitdrukking tot 'n enkele trigonometriese term: \[ \frac{\sin 10^{\circ} + \tan(360^{\circ} - \theta) \cdot \sin 20^{\circ}}{\cos 440^{\circ}} \] 6.2 Gegee: $ \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = k \cos 2x $ 6.2.1 Bereken die waarde van $ k $ as $ \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = k \cos 2x $ - NSC Mathematics - Question 6 - 2022 - Paper 2

Step 1

Vereenvoudig, sonder die gebruik van 'n sakrekenaar, die volgende uitdrukking tot 'n enkele trigonometriese term:

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Answer

First, we simplify the expression:

\frac{\sin 10^{\circ} + \tan(360^{\circ} - \theta) \cdot \sin 20^{\circ}}{\cos 440^{\circ}}

Using identities:

Since ( \cos 440^{\circ} = \cos(440^{\circ} - 360^{\circ}) = \cos 80^{\circ} ), we have:

\frac{\sin 10^{\circ} + \tan(360^{\circ} - \theta) \cdot \sin 20^{\circ}}{\cos 80^{\circ}}

Now, using the identity ( \tan(360^{\circ} - \theta) = -\tan\theta ), this results in:

\frac{\sin 10^{\circ} - \tan \theta \cdot \sin 20^{\circ}}{\cos 80^{\circ}}

Then, we apply the double angle identity:

\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}

Hence, it simplifies to:

1 - 2\sin^2\theta

Thus, we find that the final simplified term is:

\cos 2\theta

Step 2

Bereken die waarde van k as $ \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = k \cos 2x $.

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Answer

To solve for ( k ), we first apply the sine sum and difference identities:

\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)

This leads us to:

\sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = 2\sin(60^{\circ}) \cos(2x)

Substituting into our equation gives:

2 \sin(60^{\circ}) \cos(2x) = k \cos 2x

Dividing both sides by ( \cos 2x ) (assuming ( \cos 2x \neq 0 )), we get:

k = 2 \sin(60^{\circ})

Calculating values, since ( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} ):

k = \sqrt{3}

Step 3

As $ \cos x = \sqrt{t} $, bepaal, sonder die gebruik van 'n sakrekenaar, die waarde van $ \tan 60^{\circ}[\sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x)] $ in terme van $ t $.

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Answer

Using the previously obtained result for ( \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) ):

\tan 60^{\circ}[\sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x)] = \tan 60^{\circ} \cdot 2\sin(60^{\circ}) \cos(2x)

Substituting the known values:

Since ( \tan 60^{\circ} = \sqrt{3} ) and ( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} ), we have:

\tan 60^{\circ} \cdot 2 \frac{\sqrt{3}}{2} \cos(2x) = \sqrt{3}\cdot \cos(2x)

Now, using the original relationship and value of ( \cos x ):

\cos x = \sqrt{t} \Rightarrow \cos(2x) = 2\cos^{2} x - 1 = 2t - 1

Thus we find:

\tan 60^{\circ}[\sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x)] = \sqrt{3}(2t - 1)

Vereenvoudig, sonder die gebruik van 'n sakrekenaar, die volgende uitdrukking tot 'n enkele trigonometriese term:

Bereken die waarde van k as \( \sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x) = k \cos 2x \).

As \( \cos x = \sqrt{t} \), bepaal, sonder die gebruik van 'n sakrekenaar, die waarde van \( \tan 60^{\circ}[\sin(60^{\circ} + 2x) + \sin(60^{\circ} - 2x)] \) in terme van \( t \).

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