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 $\tan 12^\circ$.

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

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

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

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

Use the inverse calculator function to find the acute angle $\alpha$, expressed in degrees, minutes, and seconds, given that $\sin \alpha = 0.456$.

Determine the acute angle $\beta$ in degrees, minutes, and seconds, given that its tangent is $\tan \beta = 2.15$.

On a warehouse loading ramp, the total length of the ramp is $15 \text{ m}$ and the vertical height it covers is $2.5 \text{ m}$. Calculate the angle of inclination of the ramp with respect to the horizontal, expressing it in degrees and minutes.

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 reciprocal identities on your calculator.

Given that $\sec \alpha = 1.8$ and that $\alpha$ is an acute angle, use the calculator to find first the value of $\alpha$ in degrees, minutes, and seconds, and then calculate the value of $\sin \alpha$ rounded to three decimal places.

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

Express in radians the following angles given in degrees, 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 that the angles are in radians. Express the result to four decimal places after correctly setting your calculator mode.

Calculate the value of the acute angle $\gamma$ in radians (rounding to three decimal places) if its cotangent is $\cot \gamma = 0.65$.

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

A phone mast casts a shadow of $34.5 \text{ m}$ on 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.

An airplane is flying at a constant altitude of $10,500 \text{ m}$. The pilot observes two cities aligned in the same direction ahead of him. The nearer city is seen at an angle of depression of $34^\circ 12'$ and the further city at an angle of depression of $18^\circ 45'$. Calculate the exact distance between the two cities using the calculator, rounding to the nearest meter.

In an optical physics experiment, a ray of light passes from air into water. Snell's law states that $n_1 \times \sin \theta_1 = n_2 \times \sin \theta_2$. 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.

A civil engineer designs a banked curve for a road. The physical formula relating the banking angle $\theta$, speed $v$ (in m/s), curve radius $R$ (in m), and gravity $g = 9.8 \text{ m/s}^2$ is $\tan \theta = \frac{v^2}{R \times g}$. If a curve of radius $150 \text{ m}$ 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.

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

Evaluate the expression $K = \sin(\arccos 0.6)$ using your calculator. Next, prove algebraically using the fundamental trigonometric identities why the numerical result obtained is exact and rational.

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

A solar panel must be positioned to optimise light collection. The percentage of collection efficiency is given by the expression $E = 100 \times \cos(\alpha - 38^\circ 15')$, where $\alpha$ is the angle of inclination of the panel. If due to roof constraints 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.