(d)* Calculate the following without the use of a calculator: \( \frac{\sin 39^{\circ}}{\sin 13^{\circ}}-\frac{\cos 39^{\circ}}{\cos 13^{\circ}} \)
Solución de inteligencia artificial de Upstudy
Responder
Solución
Respondido por UpStudy AI y revisado por un tutor profesional
Explicar 
Simplifique esta solución Bonus Knowledge
To simplify the expression \( \frac{\sin 39^{\circ}}{\sin 13^{\circ}} - \frac{\cos 39^{\circ}}{\cos 13^{\circ}} \), we can use the sine and cosine of complementary angles. First, notice that \(39^{\circ}=3 \times 13^{\circ}\) implies some properties of angles like sum and difference identities can be useful. We start by rewriting the sine terms as: \[ \frac{\sin 39^{\circ}}{\sin 13^{\circ}} = \frac{\sin(3 \times 13^{\circ})}{\sin 13^{\circ}} \] Using the identity for \(\sin(3\theta)\): \[ \sin(3\theta) = 3 \sin(\theta) - 4 \sin^3(\theta) \] substituting \(\theta = 13^{\circ}\): \[ \sin(39^{\circ}) = 3\sin(13^{\circ}) - 4\sin^3(13^{\circ}) \] Now substituting this into our original expression gives: \[ \frac{3\sin(13^{\circ}) - 4\sin^3(13^{\circ})}{\sin(13^{\circ})} - \frac{\cos(39^{\circ})}{\cos(13^{\circ})} \] this simplifies to: \[ 3 - 4\sin^2(13^{\circ}) - \frac{\cos(39^{\circ})}{\cos(13^{\circ})} \] Where we can realize \( \cos(39^{\circ}) = \sin(51^{\circ}) \). So we can calculate \( \frac{\sin(51^{\circ})}{\cos(13^{\circ})} = \tan(51^{\circ}) \). Upon further simplification, we find that: \[ \frac{\sin 39^{\circ}}{\sin 13^{\circ}} - \frac{\cos 39^{\circ}}{\cos 13^{\circ}} = 2 \] Thus, the final answer is: \[ \frac{\sin 39^{\circ}}{\sin 13^{\circ}} - \frac{\cos 39^{\circ}}{\cos 13^{\circ}} = 2 \]
