Using the calculator
Aoife is walking up a ramp at the Cliffs of Moher visitor centre and wants to know how steep it is. You already know what the sine, cosine and tangent of an angle are. The problem is that, apart from a few special angles ($30^\circ$, $45^\circ$, $60^\circ$...), their exact value cannot be worked out by hand: you need a calculator, and you have to know how to use it accurately, because one wrong button press gives a result that looks correct but is completely wrong.
Direct ratios: $\sin$, $\cos$, $\tan$
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…
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. a) Without using the…
Before turning to the calculator, it is advisable to anticipate the magnitude of the result to detect possible typing mistakes or angle mode errors.
- 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 $0.65$.
- 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 $3$).
- 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.
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…
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,…
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.
Degrees, minutes and seconds
An angle does not always come in whole degrees: $37^\circ 42' 15''$ means $37$ degrees, $42$ minutes ($1^\circ = 60'$) and $15$ seconds ($1' = 60''$). The calculator has a key (usually $\boxed{^\circ \, ' \, ''}$) for entering and converting these values.
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…
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…
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.
Inverse functions: from the value of the ratio to the angle
If you know the value of $\sin \alpha$ and want to find $\alpha$, you use the inverse function: $\sin^{-1}$ (or $\arcsin$), normally with the $\boxed{\text{SHIFT}}$ or $\boxed{\text{2ndF}}$ key followed by $\boxed{\sin}$.
Use the inverse calculator function to find the acute angle $\alpha$, expressed in degrees, minutes, and seconds, given that $\sin \alpha = 0.456$.
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 $\tan \beta = 2.15$.
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…
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.
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…
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.
Reciprocal ratios: secant, cosecant and cotangent
Most scientific calculators do not have a direct key for $\sec$, $\csc$ or $\cot$: you have to calculate $\cos$, $\sin$ or $\tan$ first, and then take the reciprocal with the $\boxed{x^{-1}}$ or $\boxed{1/x}$ key.
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…
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…
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.
Degrees or radians: choosing the right mode
An angle can also be measured in radians: one complete turn is $360^\circ$ or $2\pi$ radians. The calculator has a mode for each unit (DEG / RAD), and you choose one or the other with the $\boxed{\text{MODE}}$ key (or $\boxed{\text{SHIFT}}$ + $\boxed{\text{MODE}}$, SETUP, depending on the model), checking the DEG/RAD indicator on the screen before calculating, not with an angle-conversion key.
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…
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…
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…
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.