Lesson 11 — Trigonometric ratios in the right triangle

In a right triangle, the acute angle $\theta$ has opposite leg 3 and adjacent leg 4. Compute $\sin\theta, \cos\theta, \tan\theta$.

A right triangle has legs 5 and 12. Compute the hypotenuse and $\sin, \cos, \tan$ of the angle opposite to leg 5.

Hypotenuse = 13 and leg opposite to $\theta$ equals 5. Compute $\sin\theta$ and $\cos\theta$.

Compute exact $\sin 30°$, $\cos 30°$, $\tan 30°$.

Compute exact $\sin 60°$, $\cos 60°$, $\tan 60°$.

Compute exact $\sin 45°$, $\cos 45°$, $\tan 45°$.

If $\sin\theta = \sqrt{3}/2$ and $\theta$ is acute, what is $\theta$?

If $\cos\theta = 1/2$ and $\theta$ is acute, what is $\theta$?

If $\tan\theta = 1$ and $\theta$ is acute, what is $\theta$?

In a $30°$-$60°$-$90°$ triangle with hypotenuse 10, compute the legs.

If $\sin\theta = 3/5$ and $\theta$ is acute, compute $\cos\theta$.

If $\cos\theta = 5/13$ and $\theta$ is acute, compute $\sin\theta$ and $\tan\theta$.

If $\tan\theta = 2/3$ and $\theta$ is acute, compute $\sin\theta$ and $\cos\theta$.

Verify the identity $\sin^2 30° + \cos^2 30° = 1$ using the notable values.

Show that $\tan\theta = \sin\theta / \cos\theta$ from the right-triangle definitions.

Show that $\sin(90° - \theta) = \cos\theta$ from the right triangle.

For acute $\theta$, which is greater: $\sin 60°$ or $\cos 60°$? Why?

Show that $\sin\theta < 1$ for every acute $\theta$. (Use the triangle inequality.)

For acute $\theta_1, \theta_2$ with $\theta_1 < \theta_2$: show that $\sin\theta_1 < \sin\theta_2$ (sine is increasing on $[0, 90°]$).

Show that $\tan\theta \to +\infty$ when $\theta \to 90°^-$.

In a right triangle with hypotenuse 20 cm and acute angle $35°$, compute the legs. (Use $\sin 35° \approx 0{.}574$ and $\cos 35° \approx 0{.}819$.)

Opposite leg = 6, angle $\theta = 40°$. Compute the hypotenuse. (Use $\sin 40° \approx 0{.}643$.)

Adjacent leg = 10, angle $\theta = 25°$. Compute the opposite leg. (Use $\tan 25° \approx 0{.}466$.)

Hypotenuse = 25, opposite leg = 7. Compute the angle $\theta$. (Ans: $\arcsin(7/25) \approx 16{.}26°$.)

Legs 8 and 15. Compute the two acute angles.

In an equilateral triangle of side $\ell$, compute the height using trigonometry. Compare with the result from Pythagoras.

In a square of side $\ell$, compute the diagonal using trigonometry.

Show that $\sin\theta + \cos\theta = \sqrt{2} \sin(\theta + 45°)$ using angle addition. (Preview Lesson 12.)

In a right triangle with adjacent leg $b$ and hypotenuse $c$, express $\tan\theta$ in terms of $b$ and $c$.

Compute (without calculator) $\sin 75°$. (Hint: $75° = 45° + 30°$. Use the sum formula — look it up if needed.)

A 5 m ladder is leaning against the wall forming a $70°$ angle with the ground. At what height does it touch the wall?

You are 50 m from the base of a tower. The angle of elevation to the top is $40°$. What is its height?

A plane takes off and reaches 1,500 m altitude horizontally 5 km from the runway. What is the climb angle?

A ship observes a lighthouse $200$ m tall under an angle of elevation of $3°$. How far is the ship from the lighthouse?

An accessibility ramp has a maximum slope of $5°$ (NBR 9050). To overcome 80 cm of height, what is the minimum length of the ramp?

In a solar eclipse, the Moon has angular diameter $0{.}5°$ seen from Earth. Real diameter: 3,474 km. Compute the Earth-Moon distance. (Use $\tan(0{.}25°) \approx 0{.}004363$.)

A 200 N force is applied to a body in a direction forming $30°$ with the horizontal. Compute the horizontal and vertical components of the force.

A 50 kg block is on a $20°$ ramp. What is the force parallel to the ramp that tends to make the block slide? ($g = 10$ m/s².)

The Eiffel Tower is 324 m tall. At what angle do you see the top if you are 500 m from the base?

A drone is 100 m high and detects a person on the ground at a $30°$ angle below the horizon (depression). Horizontal distance drone-person?

A surveyor is at a point $A$ and measures the top of a hill at an angle of $25°$. He walks 100 m toward the hill to $B$ and the angle becomes $40°$. Compute the height of the hill. (System of two equations.)

A steel cable supports a 30 m antenna fixed to the ground 12 m from the base. What is the length of the cable? What is the angle of the cable with the ground?

In a 1 m pendulum, the wire forms a $15°$ angle with the vertical at the extreme. What is the height of the extreme above the equilibrium point?

GPS computes your position using angles to 4 satellites. Simplified 2D model: two satellites at $(0, 20{,}000)$ km and $(15{,}000, 25{,}000)$ km see you under angles $\alpha, \beta$ with the vertical. (Sketch, do not solve — visualize triangulation.)

A road makes a V-curve with $8\%$ ascent followed by $5\%$ descent. Compute the angles of ascent and descent in degrees.

Show that in a right triangle with angles $\theta$ and $90° - \theta$, we have $\sin\theta \cdot \cos\theta = \sin(\theta) \cos(90° - \theta)/2$. (More general: $\sin(2\theta) = 2\sin\theta\cos\theta$.)

In a right triangle, show that area $= \frac{1}{2} c^2 \sin\theta \cos\theta$ where $c$ is the hypotenuse and $\theta$ one of the acute angles.

Solve: $\sin\theta + \cos\theta = 1$ for $\theta \in [0°, 90°]$.

Prove $\sin^2\theta + \cos^2\theta = 1$ using the right triangle and Pythagoras.

Show that $\sin\theta = \cos(90° - \theta)$ for every $\theta \in (0°, 90°)$.