Use P Q R with right angle R to find sinP, cosP, tanP sinQ, cosQ, and tan Q . Express each ratio

Use P Q R with right angle R to find sinP, cosP, tanP sinQ,...

Key Concepts

Trigonometric Ratios in Right Triangles

In any right triangle, the sine, cosine, and tangent functions are defined as the ratios of specific sides relative to an acute angle. Sine is the ratio of the side opposite the angle to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. These definitions allow you to compute the primary trigonometric functions once the side lengths are known.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is essential both for verifying the consistency of given side lengths and for calculating a missing side length when the other two are known.

Complementary Angles

In a right triangle, the two acute angles are complementary, meaning they add up to 90°. This property implies a relationship between their trigonometric ratios, such as the sine of one angle equaling the cosine of the other. Understanding these complementary relationships can simplify calculations and provide checks on computed values.

Conversion of Fractions to Decimals

Once the trigonometric ratios have been derived as exact fractions, they can be converted to decimal form to facilitate interpretation and comparison. This involves performing the division indicated by the fraction and, often, rounding the result to a specified precision, such as to the nearest hundredth.

Transcript

00:01 This problem wants us to use triangle pqr with right angle r to find sine of p, cosine of p, and tangent of p, as well as sine of q, cosine of q, and tangent of q.

00:09 And we are supposed to express each ratio as a fraction and as a decimal to the nearest hundredth.

00:14 So first to set up all of our fractions for all six of our problems, we need to remember what sine, cosine, and tangent represent in our right angle based off of our angle we're focused on.

00:24 And for our first three problems, since we're focused on angle p, we will find sine of p by remembering that sine is the opposite side of the hypotenuse side in comparison to p.

00:35 So in comparison to p, 12 is our opposite side and our hypotenuse is always across from our right angle.

00:41 So that means 37 is our hypotenuse.

00:44 So sine of p would be equal to 12 over 37.

00:50 And then for cosine of p, that's the adjacent of the hypotenuse.

00:54 So in comparison to p, adjacent would be the side next to it that's not the hypotenuse, which is 35, still over the same hypotenuse, 37.

01:02 And then for tangent of p, that will be the opposite side over the adjacent side.

01:07 So that's our opposite 12 over our adjacent 35.

01:12 And that sets up all of our fractions.

01:13 And now we're going to focus on our q angle for sine of q, cosine of q, and tangent of q, which means we're going to see a lot of the same sides used, but just in different combinations, because now we're focused on the q angle instead of p.

01:25 So sine is still opposite over our hypotenuse.

01:28 So opposite of q would be 35...

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