Today’s Wonder of the Day was inspired by Sai. Sai Wonders, “How do sine, cosine, and tangents work?” Thanks for WONDERing with us, Sai!

Many of our Wonder Friends already know about the Pythagorean Theorem. They may have read about Pascal’s Triangle. They might even know the other types of triangles. Today, we’re getting curious about another topic related to triangles. What are we talking about? Sine, cosine, and tangent, of course!

What are sine, cosine, and tangent? They’re three of the main functions in trigonometry. You may have heard about them already in math class. Trigonometry is connected to geometry and other branches of math. It can help us better understand the connections between the sides and angles of rectangles.

Sine, cosine, and tangent are important to the study of right triangles. Have you ever seen this type of triangle? If so, you know that one of its three angles is always 90° (a right angle). The other two angles may have any measurements, as long as all three angles add up to 180°.

How do mathematicians find sine, cosine, and tangent? Thay start with one of the non-right angles in a right triangle. Typically, they label this angle as theta (Θ). Then, they label the three sides of the triangle.

The side of the triangle directly across from the right angle is called the hypotenuse. It’s the longest side on the triangle. The side across from theta is called “opposite.” That’s pretty easy to remember—it’s on the opposite side from the theta angle! Finally, the side that touches theta but is not the hypotenuse is called “adjacent.”

Labeling the sides correctly is very important in finding sine, cosine, and tangent. That’s because each one has a formula that divides the length of one side by the length of another. Here are the formulas for finding each of these functions:

• Sine Θ = Opposite ÷ Hypotenuse

• Cosine Θ = Adjacent ÷ Hypotenuse

• Tangent Θ = Opposite ÷ Adjacent

At first glance, these formulas may seem difficult to remember. However, memorizing a useful mnemonic can help. When finding sine, cosine, and tangent, just remember SOHCAHTOA (sō-kŭ-tō-ŭ). In this example, S, C, and T stand for sine, cosine, and tangent. O, A, and H stand for Opposite, Adjacent, and Hypotenuse.

By remembering SOHCAHTOA, you’ll be able to correctly write the formulas for sine, cosine, and tangent. Then, just plug in the right numbers, and you’re set! Finding these functions is a matter of remembering the formulas and using the correct side measurements. Many people find it helpful to label the triangle before getting started.

Can you find the sine, cosine, and tangent of an angle? We bet you can! New math topics can be difficult at first, though. With practice and help from a teacher or friend, we know you’ll soon be learning lots about trigonometry!

Standards: CCSS.MATH.HSF.TF.A.2, CCRA.R.1, CCRA.R.2, CCRA.R.4, CCRA.R.10, CCRA.L.1, CCRA.L.2, CCRA.L.3, CCRA.L.6, CCRA.W.2, CCRA.W.4, CCRA.W.9, CCRA.W.10, CCRA.SL.1, CCRA.SL.2

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