Let's remind ourselves of the names of the sides of a right triangle relative to an angle
The three fundamental trigonometric ratios, sine, cosine, and tangent, represent the ratios of the sides of a right triangle. The ratios are often shortened to , and and they are defined as
For a given angle , these ratios remain the same no matter what the size of the triangle is. For example, if we were to double the length of each side of a given triangle, the values of these ratios would remain unchanged.
An easy way of remembering how to compute the different trigonometric ratios is by using the three-syllable acronym SOH-CAH-TOA. It stands for:
SOH Sine = Opposite / Hypotenuse,
CAH Cosine = Adjacent / Hypotenuse,
TOA Tangent = Opposite / Adjacent.
Find and for the triangle below.
Remember the acronym "SOH-CAH-TOA."
The sine of an angle is equal to the opposite divided by the hypotenuse (S-O-H). Relative to the angle the opposite side is and the hypotenuse is Therefore,
The cosine of an angle is equal to the adjacent divided by the hypotenuse (C-A-H). Relative to the angle the adjacent side is and the hypotenuse is Therefore,
The tangent of an angle is equal to the opposite divided by the adjacent (T-O-A). Relative to the angle the opposite side is and the adjacent side is Therefore,
Use the diagram above to determine $\cos{Q}.$
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a
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b
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c
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d
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e
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In the right triangle $\triangle KLM,$ $\angle K$ is a right angle, $KL=24,$ $KM=32$ and $LM=40.$ Find $\sin M.$
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a
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$\dfrac{2}{5}$ |
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b
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$\dfrac{3}{5}$ |
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c
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$\dfrac{4}{5}$ |
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d
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$\dfrac{1}{4}$ |
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e
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$\dfrac{3}{4}$ |
For the right triangle $PQR$ above, which of the following statements are true?
- $\cos{Q} = \dfrac{8}{17}$
- $\tan{Q} = \dfrac{8}{15}$
- $\sin{R} = \dfrac{15}{17}$
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a
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I only |
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b
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II only |
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c
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III only |
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d
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I and II only |
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e
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II and III only |
In the right triangle the angle is a right angle, and the side lengths are and Find
Let's illustrate the given triangle.
The tangent of an angle is equal to the opposite divided by the adjacent (T-O-A). Relative to the opposite side is and the adjacent side is Therefore,
Use the diagram above to determine $\sin{C}.$
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b
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c
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d
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e
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In the right triangle $\triangle ABC,$ $\angle A$ is a right angle, $AB=2\sqrt{7},$ $AC=6$ and $BC=8.$ Find $\sin B.$
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a
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$\dfrac{\sqrt 7}{3}$ |
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b
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$\dfrac{4}{3}$ |
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c
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$\dfrac{\sqrt{21}}{7}$ |
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d
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$\dfrac{\sqrt 7}{4}$ |
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e
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$\dfrac{3}{4}$ |
In the right triangle $\triangle{STU},$ $\angle S$ is a right angle, $ST=2\sqrt{10},$ $SU=3,$ and $TU=7.$ Find $\tan T.$
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b
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c
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d
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e
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Given that find the value of
The tangent of an angle is equal to the opposite divided by the adjacent (T-O-A).
Relative to the angle the length of the opposite side is and the length of the adjacent side is Therefore,
Given that $\cos\theta =\dfrac{11}{61}$, the value of $x$ is
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a
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$11$ |
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b
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$2$ |
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c
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$6$ |
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d
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$13$ |
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e
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$7$ |
If $\sin\theta = \dfrac{12}{13},$ then $x = $
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b
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c
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d
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e
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Calculators usually have a special key that allows us to compute and for some specific value of
For example, using a calculator, we can compute the trigonometric ratios for and get the following results:
Watch out! Make sure that the calculator is in degrees mode (instead of radians mode) before you start the computation.
Evaluate Round the answer to two decimal places.
Using a calculator, we find
rounded to two decimal places.
Rounded to $3$ decimal places, $\sin\left(72^\circ\right)\approx$
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a
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b
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c
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d
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e
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Rounded to $3$ decimal places, $\tan\left(55^\circ\right)=$
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a
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$0.574$ |
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b
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$1.181$ |
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c
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$0.819$ |
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d
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$1.574$ |
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e
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$1.428$ |
Which of the following is true for $\angle B?$
- $\cos{B} \approx 0.82$
- $\sin{B} \approx 0.44$
- $\tan{B} \approx 0.70$
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a
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I and III only |
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b
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III only |
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c
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I only |
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d
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II only |
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e
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II and III only |