Right Triangle Trigonometry Calculator

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Example

Created on 2024-06-20Asked by Avery Jones (Solvelet student)

Given a right triangle with an angle

\theta = 30^\circ

and the adjacent side

a = 5

, find the lengths of the opposite side

b

and the hypotenuse

c

Solution

To find the lengths of the opposite side

b

and the hypotenuse

c

for a right triangle with

\theta = 30^\circ

and adjacent side

a = 5

: 1. **Use the trigonometric functions for a 30-degree angle:**

\cos(30^\circ) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin(30^\circ) = \frac{1}{2}.

2. **Find the hypotenuse

c

:**

\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c},

\frac{\sqrt{3}}{2} = \frac{5}{c} \quad \Rightarrow \quad c = \frac{5 \cdot 2}{\sqrt{3}} = \frac{10}{\sqrt{3}} = \frac{10\sqrt{3}}{3}.

3. **Find the opposite side

b

:**

\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c},

\frac{1}{2} = \frac{b}{\frac{10\sqrt{3}}{3}} \quad \Rightarrow \quad b = \frac{1}{2} \cdot \frac{10\sqrt{3}}{3} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3}.

Therefore, the lengths of the opposite side

b

and the hypotenuse

c

are

\frac{5\sqrt{3}}{3}

and

\frac{10\sqrt{3}}{3}

respectively. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Noah Flores on Solvelet

1. Given a right triangle with an angle of 30 degrees and a hypotenuse of 10 units, find the length of the opposite side.2. Determine the measure of an acute angle in a right triangle with legs of length 3 and 4 units.,

DefinitionTrigonometry is the study of the properties of triangles with a specific emphasis on the angle measures and side lengths of right triangles. The four main functions are sine, cosine, tangent, and their reciprocal cotangent (cosine, sine, and secant are all reciprocals, and cosecant is the inverse of sine). This means that, for a right triangle with an angle θ, sin(θ)=hypotenuseopposite, cos(θ)=hypotenuseadjacent, and tan(θ)=adjacentopposite

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