We can use Greek letters such as \(\theta\) to represent an angle. We label the sides of a right triangle based on the
acute angle (\(\theta\)) of reference.
acute angle (\(\theta\)) of reference.
Notice that the hypotenuse is always the side across from the right angle. The opposite and adjacent sides change based on the location of the angle \(\theta\). The opposite side is across from the acute angle \(\theta\), and the adjacent side is next to \(\theta\).
To summarize:
Hypotenuse: side across from the right angle
Opposite: side across from the known acute angle, \(\theta\)
Adjacent: side next to the known acute angle, \(\theta\), which is not the hypotenuse (this is the only unlabeled side!)
Two right triangles with ONE other known angle congruent must be similar by \(AA\sim\).
To summarize:
Hypotenuse: side across from the right angle
Opposite: side across from the known acute angle, \(\theta\)
Adjacent: side next to the known acute angle, \(\theta\), which is not the hypotenuse (this is the only unlabeled side!)
Two right triangles with ONE other known angle congruent must be similar by \(AA\sim\).
Since \(\triangle ABC\sim\triangle DEF\), we know the sides are proportional
\(\Large\frac{b}{c}=\frac{f}{e}\;\;\;\;\) or \(\;\;\;\;\;\;\Large\frac{\text{opp}}{\text{hyp}}=\frac{\text{opp}}{\text{hyp}}\)
Right Triangle Trigonometry uses the \(AA\sim\) to determine lengths in any RIGHT triangle given one acute angle measure and one side length. [Recall: for Pythagorean Theorem, two side lengths are needed.]
\(\Large\frac{b}{c}=\frac{f}{e}\;\;\;\;\) or \(\;\;\;\;\;\;\Large\frac{\text{opp}}{\text{hyp}}=\frac{\text{opp}}{\text{hyp}}\)
Right Triangle Trigonometry uses the \(AA\sim\) to determine lengths in any RIGHT triangle given one acute angle measure and one side length. [Recall: for Pythagorean Theorem, two side lengths are needed.]
Sine, Cosine and Tangent
Mathematicians noticed that because \(AA\sim\) the ratios of sides lengths in right triangles have to be equal. They decided to give names to these trigonometric ratios.
The sine of an angle (written as \(\sin\theta\)) is defined as sin\(\theta=\Large\frac{\text {length of the side opposite}\, \theta}{\text{length of the hypotenuse}}\)
The cosine of an angle (written as \(\cos\theta\)) is defined as cos\(\theta=\Large\frac{\text{length of the side adjacent}\, \theta}{\text{length of the hypotenuse}}\)
The tangent of an angle (written as \(\tan\theta\)) is defined as tan\(\theta=\Large\frac{\text{length of the side opposite}\, \theta}{\text{length of the side adjacent}\,\theta}\)
One way to remember the trigonometric ratios to think of trig as a day at the beach, so let’s (soak a toe-a).
Mathematicians noticed that because \(AA\sim\) the ratios of sides lengths in right triangles have to be equal. They decided to give names to these trigonometric ratios.
The sine of an angle (written as \(\sin\theta\)) is defined as sin\(\theta=\Large\frac{\text {length of the side opposite}\, \theta}{\text{length of the hypotenuse}}\)
The cosine of an angle (written as \(\cos\theta\)) is defined as cos\(\theta=\Large\frac{\text{length of the side adjacent}\, \theta}{\text{length of the hypotenuse}}\)
The tangent of an angle (written as \(\tan\theta\)) is defined as tan\(\theta=\Large\frac{\text{length of the side opposite}\, \theta}{\text{length of the side adjacent}\,\theta}\)
One way to remember the trigonometric ratios to think of trig as a day at the beach, so let’s (soak a toe-a).
Three of the ratios used for Right Triangle Trig are shown above. Choose the trigonometric ratio according to the side length you are given and the side length you are trying to find.
Example 1:
Write each trig ratio. sin \(N =\) sin \(R = \) cos \(N =\) cos \(R =\) tan \(N =\) tan \(R =\) Note that the labels of “opposite” and “adjacent” will depend on if you are labeling from \(\angle N\) or \(\angle R\). It may be helpful to draw the triangle once and label it from $\angle N$, then draw the triangle again and label from \(\angle R\) or circle the angle of reference. |
Solution:
sin \(N =\Large\frac{9}{41}\)
sin \(R = \Large\frac{40}{41}\)
cos \(N =\Large\frac{40}{41}\)
cos \(R =\Large\frac{9}{41}\)
tan \(N =\Large\frac{9}{40}\)
tan \(R = \Large\frac{40}{9}\)
What do you notice about the ratio for sin \(N\) and cos \(R\)?
How about sin \(R\) and cos \(N\)?
sin \(N =\Large\frac{9}{41}\)
sin \(R = \Large\frac{40}{41}\)
cos \(N =\Large\frac{40}{41}\)
cos \(R =\Large\frac{9}{41}\)
tan \(N =\Large\frac{9}{40}\)
tan \(R = \Large\frac{40}{9}\)
What do you notice about the ratio for sin \(N\) and cos \(R\)?
How about sin \(R\) and cos \(N\)?
Cosine is short for the "complement's sine", which means the sine of the complementary angle. Recall, the measures of a pair of complementary angles adds to \(90^{\circ}\). Since angle measures in a triangle must add to \(180^{\circ}\), the two acute angles in a right triangle must add to \(90^{\circ}\), and therefore are complementary. So the cosine of \(\angle R\) is the same as the sine of \(\angle N\), since \(\angle R\) and \(\angle N\) are complementary angles.
Example 2:
Find the values of \(x\) and \(y\).
Example 2:
Find the values of \(x\) and \(y\).
Important Calculator Note:
Your calculator knows that there are TWO ways to measure angles: radians and degrees. For Right Triangle Trig, your calculator should be in DEGREE MODE. You check this in Desmos by looking at the Graph Settings, by pressing the MODE button on your calculator, or just by checking that sin \(30^{\circ} = 0.5\).
Your calculator knows that there are TWO ways to measure angles: radians and degrees. For Right Triangle Trig, your calculator should be in DEGREE MODE. You check this in Desmos by looking at the Graph Settings, by pressing the MODE button on your calculator, or just by checking that sin \(30^{\circ} = 0.5\).