8.2. Basic identities

Define for angle A in ΔABC: • sin(A) = opposite/hypotenuse = a/c • cos(A) = adjacent/hypotenuse = b/c • tan(A) = opposite/adjacent = a/b

In a right triangle, each angle and side can be related to one another using specific ratios. For an angle, like angle A, we can define three essential trigonometric functions:

1. Sine (sin) is the ratio of the length of the side opposite angle A to the length of the hypotenuse. This means that if we look across from angle A, the length of that side (let's call it 'a') divided by the hypotenuse (which is 'c') gives us sin(A).
2. Cosine (cos) is similar but involves the adjacent side. It compares the side next to angle A (let's call that side 'b') to the length of the hypotenuse. So, cos(A) = b/c.
3. Tangent (tan) is another ratio that compares the opposite side to the adjacent side, thus tan(A) = a/b. These ratios are foundational for understanding angles and sides in a right triangle.

Imagine you are standing at the base of a tall building and looking up at the top of the building. The angle at which you are looking up is angle A. The height you must look up to is the 'opposite' side (let's say it's 10 meters), the distance from the point where you stand directly below the top of the building to the building (the base) is the 'adjacent' side (say 5 meters), and the line of sight from your eyes to the top of the building is the hypotenuse. Understanding these ratios helps in scenarios like construction, where we often need to calculate heights or distances using these angles.

Basic identities: sin²A + cos²A = 1 tan(A) = sin(A) / cos(A)

These identities are fundamental truths in trigonometry that help us simplify and solve problems involving angles and side lengths in triangles. The first identity, sin²A + cos²A = 1, tells us that if we take the sine of an angle A and square it, then add the square of the cosine of the same angle, the result will always equal 1. This is a key relationship that holds true for any angle. The second identity, tan(A) = sin(A) / cos(A), means that the tangent of an angle can be found by dividing the sine of that angle by its cosine. These identities are used frequently in various calculations and proofs in mathematics.

Think of the relationship between sine, cosine, and tangent like a triangle trainer at a gym. Whenever you are working on your fitness (which in this analogy is like finding the angle or side), the trainer (the identity) reminds you to balance your workout: you can't just focus on one exercise (like sine or cosine) without acknowledging the others, hence the sum (or relationship) should equal health—represented by 1. This balance is essential for your fitness routine, similar to how these identities balance each other in trigonometry.

Used to compute missing sides/angles. Worked example: In ΔABC right-angled at C, with side AB = 13, BC = 12, find AC and sin(A). AC = √(13² − 12²) = 5. sin(A) = 5 / 13 ≈ 0.3846.

In practical applications, basic trigonometric identities help us find unknown lengths or angles in a right triangle. For instance, consider a right triangle ΔABC where we know the lengths of two sides: AB (the hypotenuse) is 13 and BC (one leg) is 12. To find the length of the other leg, AC, we use the Pythagorean theorem, which fits right with our identities: AC = √(AB² - BC²). Plugging in our values gets us AC = √(13² - 12²) = √(169 - 144) = √25 = 5. Now, to find the sine of angle A, which is sin(A) = opposite/hypotenuse (AC/AB), we can calculate: sin(A) = 5 / 13 ≈ 0.3846. This tells us about the angle A in relation to the right triangle.

Imagine you're an engineer designing a ramp that must connect two different heights. Knowing one side's lengths helps you find the height of the ramp without any guesswork. If your ramp's longest side is 13 meters (like AB) and you can measure one leg at 12 meters (like BC), calculating the missing leg (the height AC) is as simple as using the relationships we discussed. Once you know the height, you can fully design the ramp, ensuring safety and functionality. This process showcases how vital trigonometric identities are in real-life engineering tasks.

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