\( 17 \sin 17^{\circ}=a \) in terms \( 4 a \) \( =2.1 \tan 17^{\circ} \)
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To recast the equation \( 17 \sin 17^{\circ} = a \) in terms of \( 4a \), we first multiply both sides by 4. This gives us: \[ 4a = 4 \times 17 \sin 17^{\circ} = 68 \sin 17^{\circ} \] Now, we need to understand the relation with the tangent. We are provided with \( 4a = 2.1 \tan 17^{\circ} \). Therefore, we equate the two expressions for \( 4a \): \[ 68 \sin 17^{\circ} = 2.1 \tan 17^{\circ} \] Now recalling the definition of \(\tan\): \[ \tan 17^{\circ} = \frac{\sin 17^{\circ}}{\cos 17^{\circ}} \] We can replace \(\tan 17^{\circ}\) with \(\frac{\sin 17^{\circ}}{\cos 17^{\circ}}\) in the equation, which gives: \[ 68 \sin 17^{\circ} = 2.1 \times \frac{\sin 17^{\circ}}{\cos 17^{\circ}} \] Assuming \(\sin 17^{\circ} \neq 0\) (which is true as it is a known angle), we can simplify this expression by dividing both sides by \(\sin 17^{\circ}\): \[ 68 = \frac{2.1}{\cos 17^{\circ}} \] From there, we can solve for \(\cos 17^{\circ}\): \[ \cos 17^{\circ} = \frac{2.1}{68} \] And that’s how we relate \(a\) and its multiples while unveiling a little trigonometric magic along the way!