Question

Explain how to find the measures of the angles in a right triangle, given the lengths of two sides.

Answer 1

Step-by-step explanation:

STEP 1:

Find a length of third side.

Use the Pythagorean theorem:

$leg^2+leg^2=hypotenuse^2$

STEP 2:

Use the trigonometric functions:

$sine=\dfrac{opposite}{hypotenuse}\\\\cosine=\dfrac{adjacent}{hypotenuse}\\\\tangent=\dfrac{opposite}{adjacent}$

STEP 3:

Use theorem: the sum of the measures of acute angles of the right triangle is 90°.

Example:

We have the length of legs: a = 3, b = 4.

Let c = hypotenuse.

Use the Pyhagorean theorem:

$c^2=3^2+4^2\\\\c^2=9+16\\\\c^2=25\to c=\sqrt{25}\\\\c=5$

Angle between a and c (β):

Use sine. Therefore b = opposite, c = hypotenuse:

$\sin\beta=\dfrac{4}{5}=0.8$

look at the picture

$\beta\approx37^o$

$\alpha+\beta=90^o\to\alpha+37^o=90^o$      subtract 37° from both sides

$\alpha=53^o$

Answer 2

STEP 1:

To find a length of third side, use the Pythagorean Theorem: leg^2 (a^2) + leg^2 (b^2) = hypotenuse^2 (c^2).

STEP 2:

Use the trigonometric functions: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Adjacent/Opposite.

STEP 3:

Use the theorem: the sum of the measures of acute angles of the right triangle is 90°, find the angle by looking for the radians in the trigonometric table and match them with the corresponding degrees. Also the angle opposite to the hypotenuse is 90°.

Example:

We have the length of legs: a = 3, b = 4.

Let c = hypotenuse.

Use the Pythagorean theorem:

3^2 + 4^2 = c^2 => c = 5.

Angle between a and c (β):

Use sine. Therefore b = opposite, c = hypotenuse (a = adjacent):

sin(β) = Opposite/Hypotenuse 4/5 = 0.8.

To get the angle, use β = 0.8, if d = angle, sin(d) = 37° because the sin of 37° is 0.7986 which is approximately 0.8.

α + β = 90° => α + 37° = 90° => (subtract 37° from both sides) α = 53°.

So: α (adjacent angle) = 53°, β (opposite angle) = 37°, and c (theta angle or 90° angle [opposite to the hypotenuse]) = 90°.