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 to the nearest tenth of a centimeter 2 marks
6. determine the length of vx to the nearest tenth. 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 opposite 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, given an angle \(\theta\) and the adjacent side \(x\), and we want to find the opposite side \(y\), we use the formula \(y=x\tan(\theta)\).

Step3: Analyze problem 3

(a) If in a right - triangle, given the hypotenuse \(h\) and the opposite side \(o\), the angle \(\alpha=\sin^{-1}(\frac{o}{h})\). (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, and we want to find the height \(h\) it reaches on the wall, we use \(h = L\sin(\theta)\).

Step5: Analyze problem 5

(a) For a right - triangle with legs \(m\) and \(n\), the hypotenuse \(H=\sqrt{m^{2}+n^{2}}\). (b) To find the acute angle \(\beta\) opposite a side, if the side is \(m\) and the hypotenuse is \(H\), \(\beta=\sin^{-1}(\frac{m}{H})\).

Step6: Analyze problem 6

In right - triangle \(VXW\) with \(\angle V = 42^{\circ}\) and \(WX = 7.2\mathrm{cm}\), and we want to find \(VX\). We know that \(\tan(V)=\frac{WX}{VX}\), so \(VX=\frac{WX}{\tan(V)}=\frac{7.2}{\tan(42^{\circ})}\approx7.2\div0.9004\approx8.0\mathrm{cm}\).

Step7: Analyze problem 7

First, find the angle of inclination of the sun \(\theta\). Given Mandy's height \(h_1 = 5.5\mathrm{ft}\) and shadow length \(s_1 = 8\mathrm{ft}\), \(\tan(\theta)=\frac{h_1}{s_1}=\frac{5.5}{8}\). Then for a tree of height \(h_2 = 20\mathrm{ft}\), let the shadow length be \(s_2\). Since \(\tan(\theta)\) is the same[SSE Completed, Client Connection Error][LLM SSE On Failure]

Answer:

Step1: Analyze problem 1

For a right - triangle with hypotenuse \(c = 5\mathrm{cm}\) and base \(a = 3\mathrm{cm}\), first find the opposite 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, given an angle \(\theta\) and the adjacent side \(x\), and we want to find the opposite side \(y\), we use the formula \(y=x\tan(\theta)\).

Step3: Analyze problem 3

(a) If in a right - triangle, given the hypotenuse \(h\) and the opposite side \(o\), the angle \(\alpha=\sin^{-1}(\frac{o}{h})\). (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, and we want to find the height \(h\) it reaches on the wall, we use \(h = L\sin(\theta)\).

Step5: Analyze problem 5

(a) For a right - triangle with legs \(m\) and \(n\), the hypotenuse \(H=\sqrt{m^{2}+n^{2}}\). (b) To find the acute angle \(\beta\) opposite a side, if the side is \(m\) and the hypotenuse is \(H\), \(\beta=\sin^{-1}(\frac{m}{H})\).

Step6: Analyze problem 6

In right - triangle \(VXW\) with \(\angle V = 42^{\circ}\) and \(WX = 7.2\mathrm{cm}\), and we want to find \(VX\). We know that \(\tan(V)=\frac{WX}{VX}\), so \(VX=\frac{WX}{\tan(V)}=\frac{7.2}{\tan(42^{\circ})}\approx7.2\div0.9004\approx8.0\mathrm{cm}\).

Step7: Analyze problem 7

First, find the angle of inclination of the sun \(\theta\). Given Mandy's height \(h_1 = 5.5\mathrm{ft}\) and shadow length \(s_1 = 8\mathrm{ft}\), \(\tan(\theta)=\frac{h_1}{s_1}=\frac{5.5}{8}\). Then for a tree of height \(h_2 = 20\mathrm{ft}\), let the shadow length be \(s_2\). Since \(\tan(\theta)\) is the same[SSE Completed, Client Connection Error][LLM SSE On Failure]