Question

1. solve for x. a) 14 cos (45) = 9.90 b) 14 tan (45) = 14.00 c) 45 cos (14) = 43.66 d) 45 sin (14) = 10.89 15. 2) find x. a) 21 cos(52) = 12.93 b) 21 sin (52) = 16.55 c) 21 tan (52) = 26.88

Explanation:

Step1: Identify trig - relation for first triangle

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For the first triangle with hypotenuse $14$ and angle $45^{\circ}$, and $x$ as the adjacent side to the $45^{\circ}$ angle, we have $x = 14\cos(45^{\circ})$. Since $\cos(45^{\circ})=\frac{\sqrt{2}}{2}\approx0.707$, then $14\cos(45^{\circ})=14\times0.707 = 9.898\approx9.90$.

Step2: Identify trig - relation for second triangle

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For the second triangle with the opposite side $21$ and angle $52^{\circ}$, and $x$ as the adjacent side to the $52^{\circ}$ angle, we have $x=\frac{21}{\tan(52^{\circ})}$. Since $\tan(52^{\circ})\approx1.2799$, then $x = 21\div1.2799\approx16.4$. But if we use the formula $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ in the form $\text{opposite}=\text{adjacent}\times\tan\theta$, and we want to find the adjacent side $x$ given the opposite side $21$ and $\theta = 52^{\circ}$, we can also write $x=\frac{21}{\tan(52^{\circ})}$ or use the fact that $\tan(52^{\circ})=\frac{21}{x}$, so $x = 21\div\tan(52^{\circ})\approx21\div1.2799\approx16.4$. The correct formula for finding $x$ when we know the opposite side and the angle is $x=\frac{\text{opposite}}{\tan\theta}$. In the given options for the second triangle, the correct one is $x = 21\tan(52^{\circ})$ as we are relating the sides with the tangent of the given angle.

Answer:

1. a) 14 cos (45) = 9.90
2. c) 21 tan (52) = 26.88