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 $\tan 12^\circ$.

Before turning to the calculator, it is advisable to anticipate the magnitude of the result to detect possible typing mistakes or angle mode errors.

1. a) Without using the calculator, knowing that $\sin 30^\circ = 0.5$ and $\sin 45^\circ \approx 0.7071$, estimate with reasoning whether $\sin 35^\circ$ will be closer to $0.55$ or to $0.65$.
2. b) Knowing that $\tan 45^\circ = 1$ and that the tangent increases as the angle approaches $90^\circ$, estimate whether $\tan 50^\circ$ will be slightly greater than $1$ (for example, around $1.2$) or much greater (around $3$).
3. c) Now, check your estimates by calculating $\sin 35^\circ$ and $\tan 50^\circ$ with the calculator in DEG mode (rounding to four decimal places). Compare the obtained values with your predictions and argue whether your deductions were correct.

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.

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.

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.

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 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.

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 $\tan \beta = 2.15$.

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.

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.

In the Irish road network, the Road Safety Authority (RSA) uses warning signs (such as P-16a) that indicate steep slopes as a percentage. In the Wicklow Gap (Co. Wicklow), a section features a sign for a $14\%$ downhill gradient, which means that the road drops $14\text{ m}$ for every $100\text{ m}$ of horizontal progress.

1. a) Model the slope using a right-angled triangle and write the trigonometric ratio that allows finding the angle of inclination $\alpha$ of the section with respect to the horizontal.
2. b) Use the calculator in DEG mode to calculate $\alpha$ in decimal degrees and, then, express it in sexagesimal degrees, minutes, and seconds.
3. c) An automatic braking system of an imported vehicle requires the angle programmed in radians. Convert the angle $\alpha$ to radians to four decimal places by multiplying by the corresponding factor. Then check the result by setting the calculator to RAD mode and calculating the direct inverse ratio again. Explain what would happen to the calculation if a technician forgets to switch the calculator mode from RAD to DEG in future measurements.

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.

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

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

Express the final result rounded to three decimal places.