Exercises: Using the calculator

Calculate, using your calculator and rounding to four decimal places, the value of the following direct trigonometric ratios: $\sin 34^\circ$, $\cos 78^\circ$, and $\tan 12^\circ$.

Find the value of the following trigonometric ratios, expressing the result to three decimal places. Make sure to correctly enter the degrees, minutes, and seconds into your calculator: $\sin(45^\circ 12' 30'')$, $\cos(12^\circ 5')$, and $\tan(89^\circ 0' 10'')$.

Use the inverse function on the calculator to find the acute angle $\alpha$, expressed in degrees, minutes, and seconds, knowing that $\sin \alpha = 0,456$.

Determine the acute angle $\beta$ in degrees, minutes, and seconds, if its tangent is known to be $\tan \beta = 2,15$.

Calculate, rounding to three decimal places, the value of the reciprocal trigonometric ratios $\sec 55^\circ$, $\csc 20^\circ$, and $\cot 70^\circ$. Remember to use the fundamental inverse relationships on your calculator.

Express the following angles given in sexagesimal degrees in radians, using the value of $\pi$ on your calculator and rounding to three decimal places: $40^\circ$, $115^\circ$, and $250^\circ$.

Calculate the value of $\sin(1,2)$ and $\cos(0,8)$ assuming the angles are in radians. Express the result to four decimal places after correctly setting the mode on your calculator.

A surveyor measures an angle of inclination on a hillside of $14^\circ 25'$. Calculate the tangent of this angle to four decimal places to find the slope coefficient of the terrain.

In an architectural project, a roof has a slope of $28\%$. Knowing that the slope percentage is equivalent to the tangent of the angle of inclination multiplied by $100$, use the calculator to find this angle in degrees, minutes, and seconds.

Calculate the numerical value of the following trigonometric expression, rounding the final result to two decimal places:

$E = \frac{\sin 35^\circ + \cos 55^\circ}{\tan 40^\circ}$

Knowing that $\sec \alpha = 1,8$ and that $\alpha$ is an acute angle, use the calculator to first find the value of $\alpha$ in degrees, minutes, and seconds, and then calculate the value of $\sin \alpha$ rounded to three decimal places.

On a loading ramp in a warehouse, the total length of the ramp is $15 \text{ m}$ and the vertical height it overcomes is $2,5 \text{ m}$. Calculate the angle of inclination of the ramp with respect to the horizontal, expressing it in degrees and minutes.

Numerically verify the fundamental identity $\sin^2 \alpha + \cos^2 \alpha = 1$ for an angle $\alpha = 37^\circ 42' 15''$. Write down the intermediate values of the sine and cosine rounded to four decimal places before squaring and adding them.

A telephone mast casts a shadow of $34,5 \text{ m}$ on the horizontal ground when the sun is at an angle of elevation of $42^\circ 15'$. Calculate the height of the mast rounding to one decimal place.

Calculate the value of the acute angle $\gamma$ in radians (rounding to three decimal places) if its cotangent is $\cot \gamma = 0,65$.

Evaluate the following trigonometric expression assuming the angles are given in radians:

$M = \sin(1,5) \cdot \cos(0,5) - \tan(0,8)$.

Express the final result rounded to three decimal places.

In an optical physics experiment, a ray of light passes from air into water. Snell's law states that $n_1 \cdot \sin \theta_1 = n_2 \cdot \sin \theta_2$. If the refractive index of air is $n_1 = 1$, that of water is $n_2 = 1,33$, and the angle of incidence is $\theta_1 = 52^\circ 30'$, calculate the angle of refraction $\theta_2$ in degrees, minutes, and seconds.

An airplane flies at a constant altitude of $10.500 \text{ ft}$. The pilot observes two cities aligned in the same direction in front of him. The nearer city is seen with an angle of depression of $34^\circ 12'$ and the further one with an angle of depression of $18^\circ 45'$. Calculate the exact distance between the two cities using the calculator, rounding to the nearest foot.

Evaluate the expression $K = \sin(\arccos 0,6)$ with your calculator. Then, show algebraically using the fundamental trigonometric identities why the numerical result obtained is exact and rational.

A civil engineer designs a banked curve for a road. The physical formula relating the banking angle $\theta$, the speed $v$ (in ft/s), the radius of the curve $R$ (in ft), and gravity $g = 9,8 \text{ m/s}^2$ is $\tan \theta = \frac{v^2}{R \cdot g}$. If a curve with a radius of $150 \text{ m}$ is designed for a maximum speed of $90 \text{ km/h}$, calculate the banking angle $\theta$ in degrees, minutes, and seconds.

Solve the trigonometric equation $3 \sin x - 2 = 0$ using the calculator to find all possible values of the angle $x$ between $0^\circ$ and $360^\circ$. Express the final results in degrees and minutes.

For very small angles, the value of the sine approximates the value of the angle expressed in radians. Calculate the absolute error and the relative error made when approximating the value of $\sin 1^\circ$ by the value of $1^\circ$ converted to radians. Express the relative error as a percentage to two decimal places.

In a mechanical pulley system, the tension $T$ (in Newtons) of a drive belt is given by the formula $T = 500 \cdot \csc(22^\circ 30')$. Calculate the value of the tension $T$ rounding to two decimal places, detailing what equivalent operations you enter into the calculator to obtain the cosecant.

A solar panel must be oriented to optimize light capture. The percentage of capture efficiency is given by the expression $E = 100 \cdot \cos(\alpha - 38^\circ 15')$, where $\alpha$ is the angle of inclination of the panel. If due to roof restrictions the panel is installed with an inclination of $\alpha = 45^\circ 20'$, use the calculator to determine its efficiency $E$ rounded to one decimal place.