📏 trigonometry

1. مسئله (الف): اگر $\sin \alpha = \frac{3}{5}$ و زاویه $\alpha$ در ربع دوم باشد، مقدار $\cos \alpha$ و $\tan \alpha$ را بیابید. 2. فرمول‌های مورد استفاده:

1. **State the problem:** Solve the equation $$2(\sin x)^2 - 5 \cos x + 1 = 0$$ for $$x$$ in the interval $$[0, 2\pi]$$, expressing answers as multiples of $$\pi$$. 2. **Use the Py

1. **State the problem:** Solve the equation $$2 \sin t \cos t + \sin t - 2 \cos t - 1 = 0$$ for $$t$$ in the interval $$[0, 2\pi]$$, where the solutions must be multiples of $$\pi

1. **State the problem:** Jack and Jill are on opposite sides of an 88 metre deep canyon. Jack sees the trail guide at an angle of depression of 45° and Jill sees the trail guide a

1. **State the problem:** Simplify the expression $$\frac{7\cos^2 x - 7}{\sin^2 x}$$. 2. **Use the Pythagorean identity:** Recall that $$\sin^2 x + \cos^2 x = 1$$, so $$\cos^2 x =

1. **Stating the problem:** Two hikers, Alan and Britney, start at the same campsite. Alan walks 4.5 km north, and Britney walks 6.2 km on a bearing of 65° from Alan's path. We nee

1. **Problem statement:** We have a triangle formed by points A, B, and L. The ship sails from A to B due North, so AB = 3 km vertically.

1. **Problem (a):** Given a right-angled triangle with sides 3 (vertical), 4 (base), and 5 (hypotenuse), and angle $\beta$ at the bottom-right vertex, find the sum $\sin \beta + \t

1. The problem involves finding the values of $x$ and $y$ in a right triangle with angles 45° and 60°, and given side lengths. 2. We use trigonometric ratios and the Pythagorean th

1. **State the problem:** We have two right-angled triangles side by side with angles $p$ and $r$ such that $\sin p = \sin r$. We need to show that this implies the quadratic equat

1. **State the problem:** Find all values of $x$ such that $$\cos(2x) = -\frac{\sqrt{3}}{2}$$ for $$0^\circ \leq x \leq 360^\circ$$. 2. **Recall the cosine values:** The cosine fun

1. **State the problem:** We are given a point (-3, -4) and a radius $r=5$. We need to find the reference angle and the rotational angle for this point. 2. **Recall the formulas:**

1. **State the problem:** Find the sine of angle $F$ in right triangle $EFD$ where $\angle D$ is the right angle, $EF=89$ (hypotenuse), $ED=80$, and $DF=39$. 2. **Recall the formul

1. **Problem b:** Find the angle $\theta$ between the ground and the ladder given $\cos \theta = \frac{5}{20}$.\n\n2. **Formula:** Use the cosine inverse function to find the angle

1. The problem involves Dylan using a clinometer to measure the height of a tree. 2. Dylan stands 6 m from the base of the tree and measures the angle of elevation using the clinom

1. **Problem statement:** (a)(i) Explain why $\angle ABC = 96^\circ$.

1. **State the problem:** Solve the inequality $$\sin^2 x - \sin x \geq 0$$. 2. **Rewrite the inequality:** Factor the left side as a quadratic in terms of $\sin x$:

1. **State the problem:** Solve the inequality $$\sin^2 x - \sin x \geq 1$$ for real values of $x$. 2. **Rewrite the inequality:** Let $y = \sin x$. The inequality becomes:

1. **Problem Statement:** Given a unit circle centered at the origin, the point (1, 0) is rotated by an angle $\theta$ about the origin. We need to find:

1. **State the problem:** We have a right triangle with legs 33 (opposite side to angle $\theta$) and 44 (adjacent side to angle $\theta$), and hypotenuse 55. We need to find the s

1. **State the problem:** We have a right triangle with legs 33 and 44, and hypotenuse 55. We want to find the angle $\theta$ at the top-left vertex. 2. **Formula used:** To find a