Using the Calculator
With the sine, cosine, and tangent you can find the height of a pyramid or the width of a ravine without getting any closer. First you will see what they are; then, how to use them. Except for a few special angles ($30^\circ$, $45^\circ$, $60^\circ$...), their value is not calculated by hand: you need a scientific calculator. And you have to know how to use it well, because one wrong key press gives a result that looks right but is completely wrong.
Sine, Cosine, and Tangent
Take a right triangle and look at one of its acute angles, $\alpha$. With respect to that angle, each side has a name: the hypotenuse (opposite the right angle), the opposite leg (the one across from $\alpha$), and the adjacent leg (the one that forms $\alpha$ together with the hypotenuse).
Why do they depend only on the angle and not on the size of the triangle? Because all right triangles with the same angle $\alpha$ are similar (they have the same angles), and in similar triangles the ratios between corresponding sides are equal.
On 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…
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
An angle doesn't 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 $\boxed{^\circ \, ' \, ''}$) to enter and to convert 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 the angle $\alpha$, you use the inverse function: $\sin^{-1}$ (also written $\arcsin$ or "arcsine"). On the calculator you get it by pressing $\boxed{\text{SHIFT}}$ or $\boxed{\text{2ndF}}$ and then $\boxed{\sin}$. It is the key you will use most in problems: with it you go from the measurements of a triangle to its angles.
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.
Angles of Elevation and Depression
When you look up at the top of a pyramid, your line of sight makes an angle of elevation with the horizontal. When you look down, from a lookout, the angle between the horizontal and your line of sight is an angle of depression. With one of those angles and a single distance, the tangent (or the sine, or the cosine) gives you the measurement you can't reach.
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.
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…
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.
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…
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.
Reciprocal Ratios: Secant, Cosecant, and Cotangent
In Secondary Year 3 you work mostly with sine, cosine, and tangent. You will use these other three ratios a lot in high school, and here you see how they are calculated.
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 invert 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 full turn is $360^\circ$ or $2\pi$ radians. The calculator has a mode for each unit (DEG / RAD), and you have to choose it with the MODE key (or SHIFT + MODE, SETUP, depending on the model) and check the DEG/RAD indicator on the screen before calculating, not by eye. In Secondary Year 3 problems you will almost always use degrees; radians are the usual unit in high school.
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.