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Chapter 5: Problem 63

What does it mean to solve a right triangle?

Short Answer

Expert verified
To solve a right triangle means to find all the unknown measures of the sides and angles of the triangle. This is done with the use of Pythagorean theorem, trigonometric ratios like sine, cosine and tangent, and their inverse functions, if necessary.

Step by step solution

01

Identifying Known and Unknowns

The first step in solving a right triangle is to identify what is known and what needs to be found. For example, you might know the length of one side and one angle, and you're trying to find the length of another side.
02

Apply Trigonometric Ratios

Depending on what is known and what needs to be found, determine the right trigonometric ratio to use. This could be sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), or tangent (opposite/adjacent). Plug in the known values to get an equation and solve it.
03

Use Pythagorean theorem

In case two sides of a right triangle are known and one side needs to be found, use Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
04

Finding Angles

If the measures of the sides are known and the measures of the angles need to be found, use inverse trigonometric functions. For example, to find an angle, use the appropriate ratio, set it equal to the known ratio of two sides, and solve for the angle using an inverse trigonometric function.

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Most popular questions from this chapter

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \sec \left(\sin ^{-1} \frac{x}{\sqrt{x^{2}+4}}\right) $$

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \cot \left(\tan ^{-1} \frac{x}{\sqrt{2}}\right) $$

Have you ever noticed that we use the vocabulary of angles in everyday speech? Here is an example: My opinion about art museums took a \(180^{\circ}\) turn after visiting the San Francisco Museum of Modern Art. Explain what this means. Then give another example of the vocabulary of angles in everyday use.

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \sec \left(\cos ^{-1} \frac{1}{x}\right) $$

Explain what is meant by one radian.
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