Using a Calculator for Trigonometric Ratios
A wheelchair ramp may rise at most 1 foot for every 12 feet of horizontal run. What angle does that ramp make with the ground? You know what sine, cosine, and tangent are, but except for a few special angles ($30^\circ$, $45^\circ$, $60^\circ$) you cannot find their values by hand. You need a calculator, and you need to use it carefully. One wrong mode or key gives an answer that looks reasonable and 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 $\tan 12^\circ$.
Calculate the numerical value of the following trigonometric expression, rounding the final result to two decimal places:…
Calculate the numerical value of the following trigonometric expression, rounding the final result to two decimal places:
$E = \frac{\sin 35^\circ + \cos 55^\circ}{\tan 40^\circ}$
Degrees, minutes, and seconds
Angles are not always whole degrees. The angle $37^\circ 42' 15''$ means $37$ degrees, $42$ minutes, and $15$ seconds, where $1^\circ = 60'$ and $1' = 60''$. This is the DMS form, and surveyors, pilots, and navigators use it all the time. Your calculator has an ANGLE menu (on a TI-84, 2nd then APPS) with the symbols $^\circ$, $'$, $''$ to enter such angles, and a $\blacktriangleright$DMS command that converts a decimal angle into DMS.
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…
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 $\tan(89^\circ 0' 10'')$.
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…
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.
Inverse functions: from the ratio to the angle
If you know the value of $\sin\alpha$ and want the angle $\alpha$, use the inverse function $\sin^{-1}$ (also written $\arcsin$). It is the second function of the same key: press 2nd, then $\boxed{\sin}$. The same works for $\cos^{-1}$ and $\tan^{-1}$. The calculator returns an acute angle when the ratio is positive.
To get the answer in degrees, minutes, and seconds, finish with the $\blacktriangleright$DMS command.
Now you can answer the opening question. The ramp rises $1$ ft over a run of $12$ ft, so $\tan\theta = \dfrac{1}{12}$. Then $\theta = \tan^{-1}\!\left(\dfrac{1}{12}\right) \approx 4.76^\circ \approx 4^\circ 46'$.
Use the inverse function on the calculator to find the acute angle $\alpha$, expressed in degrees, minutes, and seconds, knowing that $\sin \alpha = 0,456$.
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 $\tan \beta = 2,15$.
Determine the acute angle $\beta$ in degrees, minutes, and seconds, if its tangent is known to be $\tan \beta = 2,15$.
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…
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.
Reciprocal ratios: secant, cosecant, cotangent
Most calculators have no key for $\sec$, $\csc$, or $\cot$. Find $\cos$, $\sin$, or $\tan$ first, then take the reciprocal with the $\boxed{x^{-1}}$ key. You can also type it as a division: 1/sin(40). Notice that the key sequence uses $\sin$ to get $\csc$, and $\cos$ to get $\sec$. Be careful not to mix them up.
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 inverse relationships on your calculator.
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…
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.
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…
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.
Degrees or radians: choose the right mode
Angles can also be measured in radians. One full turn is $360^\circ$, or $2\pi$ radians. The calculator has one mode for each unit, and you must convert between them with a formula. Don't just change the mode and hope.
When the calculator is in Radian mode, the inverse functions also answer in radians. If you want an angle in degrees, switch back to Degree mode first (or convert the result with $\dfrac{180}{\pi}$).
Express the following angles given in sexagesimal degrees in radians, using the value of $\pi$ on your calculator and rounding to three decimal places: $40^\circ$,…
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…
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.
Calculate the value of the acute angle $\gamma$ in radians (rounding to three decimal places) if its cotangent is $\cot \gamma = 0,65$.
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…
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.
Solving right triangles with a calculator
The calculator is the tool that turns right-triangle trigonometry into numbers. Recall SOH-CAH-TOA:
$$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$$To find a side, you know an angle: write the ratio, solve for the side, then use $\sin$, $\cos$, or $\tan$. To find an angle, you know two sides: write the ratio, then use $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$. The angle of elevation is measured up from the horizontal, and the angle of depression is measured down from the horizontal. Both are measured from a horizontal line.
Some problems need two triangles. Here the plane sees two cities in the same direction, at two different angles of depression. The angle of elevation at each city is equal to the angle of depression from the plane, so the angles are labeled at the cities.
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…
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.
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…
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.
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…
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.
Two angles, one value: solving $\sin x = k$
Suppose you must find every angle $x$ between $0^\circ$ and $360^\circ$ with $\sin x = 0.4$. The calculator gives only one: $\sin^{-1}(0.4) \approx 23.58^\circ$. But the unit circle shows a second angle with the same sine. The sine is the height of the point on the circle, and the height $0.4$ is reached twice, once in Quadrant I and once in Quadrant II.
A related idea: sometimes the calculator's answer is exact even though it looks like a decimal. For example, the expression $\sin(\cos^{-1} 0.8)$ means "the sine of the angle whose cosine is $0.8$". The calculator shows exactly $0.6$. Algebra tells you why. Let $\alpha = \cos^{-1}(0.8)$, so $\cos\alpha = 0.8$ with $\alpha$ acute. By the identity $\sin^2\alpha + \cos^2\alpha = 1$, we get $\sin\alpha = \sqrt{1 - 0.8^2} = \sqrt{0.36} = 0.6$. The positive root is the right one because $\alpha$ is acute.
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.
Evaluate the expression $K = \sin(\arccos 0,6)$ with your calculator. Then, show algebraically using the fundamental trigonometric identities why the numerical result…
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.
Checking values and measuring error
A calculator gives decimals that have been rounded, so small differences are normal. You can test your results with the identity $\sin^2\alpha + \cos^2\alpha = 1$. Find $\sin\alpha$ and $\cos\alpha$, round them as the problem says, square them, and add. The sum should be very close to $1$, though rounding can make it come out as $0.9999$ or $1.0001$.
When an approximation is used instead of an exact value, we measure how good it is with two kinds of error:
For small angles, $\sin\theta$ is close to $\theta$ when $\theta$ is in radians. Let's test this with $5^\circ$.
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…
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.
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…
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.
Using trig in science and engineering formulas
Formulas in physics and engineering often contain a trig function. The procedure is always the same: substitute the numbers, work out what is inside the parentheses first, then use the calculator, in the right mode, without rounding in between.
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…
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.
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…
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.
A solar panel must be oriented to optimize light capture. The percentage of capture efficiency is given by the expression $E = 100 \cdot \cos(\alpha - 38^\circ 15')$,…
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.