Solutions

14. Prove:

$\cos x + \tan x \sin x = \sec x$ Proof:

1. Rewrite $\tan x$ as $\frac{\sin x}{\cos x}$: $\cos x + \tan x \sin x = \cos x + \frac{\sin x \cdot \sin x}{\cos x}$
2. Simplify the second term: $\cos x + \frac{\sin^2 x}{\cos x} = \frac{\cos^2 x}{\cos x} + \frac{\sin^2 x}{\cos x}$
3. Combine terms under a common denominator: $\frac{\cos^2 x + \sin^2 x}{\cos x}$
4. Use the Pythagorean identity $\cos^2 x + \sin^2 x = 1$: $\frac{1}{\cos x} = \sec x$ Thus, $\cos x + \tan x \sin x = \sec x$.

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15. Simplify:

$\cos \frac{\pi}{4} \cos \frac{\pi}{6} + \sin \frac{\pi}{4} \sin \frac{\pi}{6}$ Use the angle addition formula: $\cos a \cos b + \sin a \sin b = \cos (a - b)$

1. Substituting $a = \frac{\pi}{4}$ and $b = \frac{\pi}{6}$: $\cos \frac{\pi}{4} \cos \frac{\pi}{6} + \sin \frac{\pi}{4} \sin \frac{\pi}{6} = \cos \left(\frac{\pi}{4} - \frac{\pi}{6}\right)$
2. Simplify the angle: $\frac{\pi}{4} - \frac{\pi}{6} = \frac{3\pi}{12} - \frac{2\pi}{12} = \frac{\pi}{12}$
3. The result: $\cos \frac{\pi}{12}$

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16. Express $\sin 3x \sin 5x$ as a sum of trigonometric functions.

Use the product-to-sum formula: $\sin A \sin B = \frac{1}{2} \left[\cos (A - B) - \cos (A + B)\right]$

1. Substituting $A = 3x$ and $B = 5x$: $\sin 3x \sin 5x = \frac{1}{2} \left[\cos (3x - 5x) - \cos (3x + 5x)\right]$
2. Simplify the angles: $\sin 3x \sin 5x = \frac{1}{2} \left[\cos (-2x) - \cos (8x)\right]$
3. Use the property $\cos(-\theta) = \cos(\theta)$: $\sin 3x \sin 5x = \frac{1}{2} \left[\cos (2x) - \cos (8x)\right]$

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17. Air traffic problem:

a) Find the distance between the planes.

Use the law of cosines: $c^2 = a^2 + b^2 - 2ab\cos C$ Here:

• $a = 30$, $b = 35$, and $C = 20^\circ$.

1. Substitute values: $c^2 = 30^2 + 35^2 - 2(30)(35)\cos 20^\circ$
2. Simplify: $c^2 = 900 + 1225 - 2100\cos 20^\circ$
3. Approximate: $c^2 \approx 2125 - 2100(0.9397) \approx 2125 - 1973.37 \approx 151.63$
4. Take the square root: $c \approx \sqrt{151.63} \approx 12.32 \, \text{miles.}$

b) Find the angle of descent for the 30-mile-out plane.

Use: $\theta = \arctan\left(\frac{\text{altitude}}{\text{distance}}\right)$

1. Altitude = $10,000 \, \text{ft} = \frac{10,000}{5280} \, \text{miles} \approx 1.89 \, \text{miles}$.
2. Distance = $30 \, \text{miles}$.
3. Substitute: $\theta = \arctan\left(\frac{1.89}{30}\right)$
4. Approximate: $\theta \approx \arctan(0.063) \approx 3.61^\circ.$

c) Find the angle between the lines of sight for the 35-mile-out pilot.

The angle between the lines of sight is the same as the interior angle at the airport. Thus, it is $20^\circ$.

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Relative Questions

1. Can you apply the product-to-sum formula to other trigonometric identities like $\cos A \cos B$?
2. How does the law of cosines apply to more complex triangles in navigation?
3. What is the significance of expressing products of trigonometric functions as sums?
4. How would the solution to part (b) change if the altitude were halved?
5. What are some real-world applications of the angle addition formula in engineering?

Tip:

Always remember to simplify complex trigonometric expressions by leveraging known formulas to reduce computation effort.