5️⃣ A ladder leans against a wall making an angle of 60° with the ground. If the foot of the ladder is 4.6 m away from the wall, find the length of the ladder.
Step-by-Step Explanation
Let the length of the ladder be $L$.
We have $\cos(60°) = \frac{\text{distance from wall}}{\text{length of ladder}} = \frac{4.6}{L}$.
Since $\cos(60°) = \frac{1}{2}$, we have $\frac{1}{2} = \frac{4.6}{L}$.
Therefore, $L = 2 \times 4.6 = 9.2$ m.
Answer: 9.2 m ✅
6️⃣ If $\sin(x) + \cos(x) = \sqrt{2}$, find the value of $\tan(x)$.
Step-by-Step Explanation
Square both sides of the equation: $(\sin(x) + \cos(x))^2 = (\sqrt{2})^2$.
This gives $\sin^2(x) + \cos^2(x) + 2\sin(x)\cos(x) = 2$.
Since $\sin^2(x) + \cos^2(x) = 1$, we have $1 + 2\sin(x)\cos(x) = 2$.
Thus, $2\sin(x)\cos(x) = 1$, or $\sin(x)\cos(x) = \frac{1}{2}$.
Assuming $x=45°$, we find $\tan(x) = \tan(45°) = 1$.
Answer: 1 ✅
7️⃣ From the top of a cliff 20 m high, the angle of depression of a boat is 30°. Find the distance of the boat from the foot of the cliff.
Step-by-Step Explanation
Let the distance of the boat from the foot of the cliff be $d$.
The angle of depression is equal to the angle of elevation.
We have $\tan(30°) = \frac{\text{height of cliff}}{\text{distance of boat}} = \frac{20}{d}$.
Since $\tan(30°) = \frac{1}{\sqrt{3}}$, we have $\frac{1}{\sqrt{3}} = \frac{20}{d}$.
Therefore, $d = 20\sqrt{3}$ m.
Answer: 20√3 m ✅
8️⃣ If $\sin(\theta) + \cos(\theta) = p$, then find the value of $\sin^3(\theta) + \cos^3(\theta)$ in terms of $p$.
Step-by-Step Explanation
Square the given equation: $(\sin(\theta) + \cos(\theta))^2 = p^2$.
This gives $\sin^2(\theta) + \cos^2(\theta) + 2\sin(\theta)\cos(\theta) = p^2$.
Since $\sin^2(\theta) + \cos^2(\theta) = 1$, we have $1 + 2\sin(\theta)\cos(\theta) = p^2$.
Thus, $2\sin(\theta)\cos(\theta) = p^2 - 1$.
We know that $\sin^3(\theta) + \cos^3(\theta) = (\sin(\theta) + \cos(\theta))(\sin^2(\theta) - \sin(\theta)\cos(\theta) + \cos^2(\theta))$.
This simplifies to $p(1 - \frac{p^2 - 1}{2}) = p(\frac{3 - p^2}{2})$.
Answer: p(3 - p^2)/2 ✅
9️⃣ If tan(θ) = 3/4, then find the value of (sin(θ) + cos(θ)).
Step-by-Step Explanation
If tan(θ) = 3/4, then sin(θ) = 3/5 and cos(θ) = 4/5.