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 you correctly enter degrees, minutes, and seconds into your calculator: $\sin(45^\circ 12' 30'')$, $\cos(12^\circ 5')$, and $\tan(89^\circ 0' 10'')$.

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 $\tan \beta = 2.15$.

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.

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.

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.

Aoife wants to calculate the cosecant of $42^\circ$ for a trigonometry problem. To do so, she presses the combination $\boxed{\text{SHIFT}}\,\boxed{\sin}$ on her calculator (which activates the $\sin^{-1}$ function) and then types $42$. Upon pressing equals, the screen displays a "Math ERROR" message. Aoife concludes that the cosecant of $42^\circ$ does not exist.

1. a) Explain with reasons the two conceptual errors Aoife has made: why the calculator gives an error when entering $\sin^{-1}(42)$ and why that key cannot be used to calculate the cosecant.
2. b) Write the exact sequence of keys she must press on a standard calculator to obtain $\csc 42^\circ$ and calculate its value rounded to four decimal places, ensuring that the calculator is in the appropriate angle mode.

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.

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.

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

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

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.

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.

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.

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.

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 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.

A surveying team reviews the operation history of a field calculator and finds only the last recorded calculation: the screen displays the result of having applied $\boxed{\tan^{-1}}\,(0,4877)$. It is known that this calculation was used to determine the angle of inclination $\alpha$ of a straight ramp that overcomes an exact vertical drop of $15\text{ m}$.

1. a) Set up the trigonometric equation relating this value to the legs of the triangle and find the horizontal length of the base of the ramp.
2. b) Determine the angle $\alpha$ with the calculator and express it both in decimal degrees (to two decimal places) and in sexagesimal degrees, minutes, and seconds using the $\boxed{^\circ\,'\,''}$ key.
3. c) If the technical report requires expressing the angle in radians, calculate this value in two ways: first algebraically converting the result using the proportion with $\pi$, and second directly calculating $\tan^{-1}(0,4877)$ after setting the calculator to RAD mode. Verify that both methods yield the same result.

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.

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.

An aeroplane 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 metre.

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.

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.

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.

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.

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.

To calculate the length of a steel guy wire on a communications antenna, it is necessary to evaluate the cosecant of the elevation angle $\alpha = 36^\circ 24' 45''$. Three 4th Year students propose different procedures using the calculator:

• Cian's method: He mentally converts the minutes and seconds to decimal, approximating to two decimal places ($24' \approx 0.40^\circ$ and $45'' \approx 0.01^\circ$, adding up to $36.41^\circ$), calculates $\sin(36.41^\circ)$, and presses the $\boxed{x^{-1}}$ key.

• Aoife's method: She enters the complete angle using the sexagesimal key $\boxed{^\circ\,'\,''}$ by typing $36\,\boxed{^\circ\,'\,''}\,24\,\boxed{^\circ\,'\,''}\,45\,\boxed{^\circ\,'\,''}$, directly calculates its sine, and then presses $\boxed{x^{-1}}$.

• Niamh's method: She converts the angle to radians on paper using the exact formula $\alpha_{\text{rad}} = \left(36 + \frac{24}{60} + \frac{45}{3600}\right) \times \frac{\pi}{180}$, enters that numerical value with the calculator in RAD mode, calculates the sine, and raises it to $-1$.

1. a) Execute the three methods on your calculator and write down the result of $\csc \alpha$ for each one to five decimal places.
2. b) Considering Aoife's result as the exact reference of the instrument, calculate the absolute error that Cian makes due to the prior rounding.
3. c) Compare the three methods and justify which is the most efficient and reliable in practice, arguing why Cian's procedure is ill-advised in technical projects.

During a practical test with a scientific calculator, Cian types $\cos(60)$ expecting to obtain $0.5$, but the screen shows $-0.9524$. Puzzled, he tries typing $\sin(30)$ and obtains $-0.9880$.

1. a) Identify the configuration error in Cian's calculator. Explain what the machine actually calculated when entering the number $60$, and justify geometrically why that value turns out to be negative based on the fact that a full turn is $2\pi \approx 6.28\text{ rad}$.
2. b) Suppose Cian's keypad has a broken mode-switching key and he must finish the exam with the machine stuck in that mode. Deduce and write the exact mathematical expression he must type for the calculator to display the correct value of $\cos(60^\circ)$. Verify by typing said expression that the result obtained is exactly $0.5$.
3. c) His classmate Aoife needs to find an acute angle $\beta$ whose sine is $0.5$. She enters $\boxed{\sin^{-1}}\,(0,5)$ into her own calculator, and the screen displays $0.5236$. Determine with reasoning which mode Aoife's calculator is in, and deduce which exact fraction of $\pi$ and how many sexagesimal degrees that value represents.