Question

1. solve for sin(b), cos(b), tan(c), find the angle b and angle c. 5 marks
2. in a right triangle, angle and the side adjacent to is. find the length of the side opposite. 2 marks
3. a right triangle has hypotenuse and the side opposite angle is. (a) find angle. (b) find the remaining side length. 2 marks
4. a ladder leans against a wall making a angle with the ground. the ladder is long. how high up the wall does it reach? 2 marks
5. a right triangle has legs and. (a) find the hypotenuse. (b) find the acute angle opposite the side 2 marks
6. determine the length of vx to the nearest tenth of a centimeter

7.2 cm
w
2 marks
mandy is standing outside in the sunshine. she is 5.5 ft tall and casts an 8ft shadow. how long of a shadow will a tree that is 20ft high make at this same time of day? after determining the angle of inclination of the sun, use the angle to determine the length of the shadow cast by the tree. 3 marks

Explanation:

Step1: Analyze problem 1

For a right - triangle with hypotenuse \(c = 5\mathrm{cm}\) and base \(a = 3\mathrm{cm}\), first find the other side \(b\) using the Pythagorean theorem \(b=\sqrt{c^{2}-a^{2}}=\sqrt{25 - 9}=\sqrt{16}=4\mathrm{cm}\). Then \(\sin(B)=\frac{b}{c}=\frac{4}{5}\), \(\cos(B)=\frac{a}{c}=\frac{3}{5}\), \(\tan(C)=\frac{a}{b}=\frac{3}{4}\). To find angle \(B\), \(B=\sin^{- 1}(\frac{4}{5})\approx53.13^{\circ}\), and angle \(C = 90^{\circ}-B\approx36.87^{\circ}\).

Step2: Analyze problem 2

If in a right - triangle, we know an angle \(\theta\) and the adjacent side \(x\), the opposite side \(y=x\tan\theta\).

Step3: Analyze problem 3

(a) If we know the hypotenuse \(h\) and the opposite side \(o\) of an angle \(\alpha\) in a right - triangle, \(\alpha=\sin^{-1}(\frac{o}{h})\). (b) The remaining side \(s=\sqrt{h^{2}-o^{2}}\) using the Pythagorean theorem.

Step4: Analyze problem 4

If a ladder of length \(L\) makes an angle \(\theta\) with the ground, the height \(h\) it reaches on the wall is \(h = L\sin\theta\).

Step5: Analyze problem 5

(a) For a right - triangle with legs \(a\) and \(b\), the hypotenuse \(c=\sqrt{a^{2}+b^{2}}\). (b) To find the acute angle \(\theta\) opposite side \(a\), \(\theta=\tan^{-1}(\frac{a}{b})\).

Step6: Analyze problem 6

In right - triangle \(VXW\) with \(\angle V = 42^{\circ}\) and \(WX = 7.2\mathrm{cm}\), \(\tan V=\frac{WX}{VX}\), so \(VX=\frac{WX}{\tan V}=\frac{7.2}{\tan42^{\circ}}\approx8.0\mathrm{cm}\).

Step7: Analyze problem 7

First, find the angle of inclination of the sun \(\theta\). Since for Mandy, \(\tan\theta=\frac{5.5}{8}\), \(\theta=\tan^{-1}(\frac{5.5}{8})\approx34.5^{\circ}\). For a tree of height \(H = 20\mathrm{ft}\), if the length of the shadow is \(S\), \(\tan\theta=\frac{H}{S}\), so \(S=\frac{H}{\tan\theta}=\frac{20}{\tan34.5^{\circ}}\approx28.8\mathrm{ft}\).

Answer:

The solutions to each problem are as described in the steps above for their respective geometric and trigonometric calculations. For example, in problem 1: \(\sin(B)=\frac{4}{5}\), \(\cos(B)=\frac{3}{5}\), \(\tan(C)=\frac{3}{4}\), \(B\approx53.13^{\circ}\), \(C\approx36.87^{\circ}\); in problem 6, \(VX\approx8.0\mathrm{cm}\); in problem 7, the length of the tree's shadow is approximately \(28.8\mathrm{ft}\).