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A stork is carrying a baby to its parents. The stork is flying at a constant altitude of 200 meters, directly approaching a delivery point that's 500 meters away horizontally, and it's about to begin its descent down to the baby's new home. The stork, being a bird, is not worried about its distance or angle of descent -- but the baby's parents, watching from below as a large wading bird with a human baby in its beak approaches solid ground, are quite concerned about the logistics of this situation. They're wondering how far the bird is from them, and more importantly, the angle at which it will need to descend to deliver their newborn to them (figuring that a smoother descent will probably be better).
Try to picture this situation and imagine what the answers to the parents' questions might be. You might not be able to answer the concerned parents now, but at the end of this section, you'll have all the tools you need to put their minds at ease. Cle
You're probably familiar with the formula for the area of a triangle and the Pythagorean Theorem already, but here are some reminders in case it's been a while since you've used these equations:
Area of triangle equals ½bh
The area of a triangle is given by
where b is the base and h is the height.
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Pythagoras' Theorem
In a right angled triangle with sides a, b and hypotenuse (longest side) c, Pythagoras' Theorem states
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Example
The right triangle ABC has [AB]=[BC]=2. Find [AC].
Since [AB]=[BC], these must be the short sides. Thus
So [AC]=√8.
Checkpoint
A right-angled triangle has a hypotenuse of length 10 and sides of length x and y.
Find the values of x and y for which the area of the triangle is 24.
Select the correct option
The Pythagorean Theorem works because it describes the relationship between the sides of a right triangle. The Pythagorean Theorem focuses on side lengths, but it's clear there's something going on with angles and as well, since the theorem relies explicitly upon the triangle being a right triangle. Let's try to translate the Pythagorean Theorem into a version that deals with angles instead of sides by investigating some similar triangles.
Discussion
The diagram shows a right triangle with an angle labeled θ. The sides are labeled A (for "adjacent" to θ), O (for "opposite" θ), and H (for "hypotenuse").
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We know from the Pythagorean Theorem that
If the hypotenuse H has length 10 and the adjacent side A has length 5√2=7.07, what is the length of the opposite side O? What are the approximate values of the ratios
HO?
HA?
AO?
If the hypotenuse H instead has length 1 and the angle θ is the same as it was in part (a), will the ratios you calculated above stay the same? Why or why not? (Remember that any two right-angled triangles with the same acute angle θ are similar)
Can you use these ratios to find the lengths of A and O in this new triangle?
Part (a)
By the Pythagorean theorem,
We are given H=10 and A=5√2 (which is approximately 7.07). You can use either the exact value or the decimal.
To find O, rearrange the formula:
Substituting the exact value:
Alternatively, using the decimal value:
Now, calculate the ratios (using either 5√2 or 7.07 for both O and A):
Part (b)
The key fact is that any two right-angled triangles with the same acute angle θ are similar, so all ratios of corresponding sides remain the same. In particular, the values of
do not depend on the size of the triangle, only on θ.
From part (a) we found (exactly)
When H=1, we use these ratios to get
Thus in the smaller triangle with hypotenuse 1, the adjacent and opposite sides both have length 0.707=2√2.
Notice how, in general, as long as the angles θ and β remain the same, changing the length of side x keeps the ratios the same. This is because in order for the angles to actually remain the same, when x is changed, the other sides must "compensate" somehow in order for the shape to stay a triangle (since all the angles of a triangle add up to 180°), which keeps the ratios of the side lengths the same. You can drag the slider around on the diagram below to get a good idea of this for yourself.
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The idea we've been alluding to in this section is trigonometric ratios. These ratios apply to right triangles and make use of fixed ratios to find unknown values. Trig ratios have many useful applications which we'll discuss shortly, but we need to first define them before going over cases where they're relevant:
Trigonometric Ratios
In a right angled triangle with an angle θ<90°, the trigonometric ratios sin, cos and tan are defined by
where opposite and adjacent refer to the side lengths of the sides opposite and adjacent to θ, while hypotenuse is the length of the longest side.
To remember these ratios, you can use the abbreviation sohcahtoa:
Example
Find cosθ in the triangle below.
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From sohcahtoa we know cosθ=ha. We can see in the diagram that o=4 and h=5. To find a, we use Pythagoras' Theorem:
So cosθ=53.
These ideas can seem pretty abstract at first, but think about what they're actually describing. In the example above, cosθ=53. This is just telling us that θ is the angle whose "cosine" is 53. And cosine is just a function we construct based on what we know about right triangles, one where if we actually constructed a right triangle, if we took the "cosine" of an acute angle, we would get the observed ratio of the sides. Basically, humans noticed how ratios and angles behaved in real life, then constructed these abstract trigonometric ratios so that they would the answers they already knew were true. The heavy lifting's already been done; we just have to plug these values into a calculator to use them now.
Try changing the side lengths of the triangle below to see how sin,cos, and tan are defined for different values.
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Exercise
The diagram shows a right triangle with side lengths 3, 2, and √13, and acute angle θ.
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Find tanθ.
Select the correct option
Now that we know the meanings of these ratios, we can begin applying them.
We saw above that if the acute angles of a right triangle remain the same, the ratios between its side lengths will as well. Clearly, the angle is tied directly to the ratio of side lengths. We can use this fact to go from a ratio to an angle.
Another way of framing this idea is: If the "sine" of θ is 21, then θ is whatever angle makes that fact true. The so-called inverse trigonometric functions will help us reverse engineer to discover the value of θ.
Discussion
We saw above that if the acute angles of a right triangle remain the same, the ratios between its side lengths will as well. Clearly, the angle is tied directly to the ratio of side lengths.
Can we use this fact to go from a ratio of sides to an angle? How do you think we would set up this process?
When two right triangles have the same acute angles, all corresponding side‐length ratios are equal. We capture those ratios in three names:
Even though these look “abstract,” each is just a real‐world ratio of two sides. We don’t need actual numbers yet—knowing the shape of the triangle fixes those ratios.
From a ratio to an angle Suppose you know the ratio of opposite : adjacent is R. By definition
so we want the angle whose tangent gives R. To do this we introduce an inverse operation to “undo” the tangent. We call this inverse function arctan (or tan−1), defined so that
gives exactly the angle whose tangent is R.
Similarly, if you know
and if
In every case the steps are:
Identify which two sides (or their ratio) you know.
Match that ratio with sin, cos or tan.
Apply the corresponding inverse function to recover the angle θ.
Finding angles in right angled triangles
If we know the value of sinθ, cosθ or tanθ in a right angled triangle, we can find θ using an inverse trigonometric function on a calculator. These functions are sin−1, cos−1 and tan−1 and satisfy
whenever θ<90°, which is always true in a right angled triangle.
Example
Find θ in the triangle below.
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The side opposite θ has length 4, and the hypotenuse has length 5. So
(the value from your GCD).
Checkpoint
The diagram below shows a right-angled triangle.
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Find the value of θ.
Select the correct option
Discussion
We've gone from ratio to angle with inverse trig ratios, but how about going from an angle and a side length to the other side length? Could we use the trig identities to go that way? How do you think we would set up this process?
We have seen that for any fixed acute angle θ in a right-angled triangle, the ratio of two of its sides is always the same. In fact we can turn each of these “constant ratios” into a little machine (a function) that, when you feed it the angle, spits out that ratio. To find an unknown side when you know an angle and one side, the steps are:
Decide which two sides you need to relate.
Pick the matching ratio-function of θ:
Plug in your known angle for θ to get a number (e.g.\ tan(30∘)=√31).
Write the equation, then solve it for the unknown side by “undoing” the division—i.e.\ multiply both sides by the known side.
For example, suppose you know an angle θ and the length of the adjacent side b, and want the opposite side a. You use
so
If instead you know the hypotenuse c and want the opposite side, you use
Similarly
In every case the pattern is the same: pick the right ratio, plug in the angle, then multiply by the known side to get the unknown one.
Finding side lengths from an angle
The trigonometric ratios sin, cos and tan are actually functions that relate an angle θ to a ratio of sides. The values of sin, cos and tan for specific angles can be found on the calculator. For example
Example
Find the length x in triangle below.
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The side with length x is opposite the angle 35°, and the hypotenuse is 5cm. Thus
Checkpoint
The diagram below shows a right-angled triangle.
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Find the length of the side x.
Select the correct option
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