What does arcsin mean?

Arcsin gives the angle whose sine is a given number, so it runs sine backwards. Since sin 30° = 0.5, arcsin 0.5 = 30°. Calculators often label it sin⁻¹, but that small −1 is not an exponent: sin⁻¹ 0.5 is an angle, not 1 divided by sin 0.5. Arccos and arctan undo cosine and tangent in the same way.

How do you find a missing side with sine or cosine?

Multiply the hypotenuse by the ratio that belongs to the side you want. Sine goes with the opposite side, and cosine with the adjacent one. If cos θ = 0.8 and the hypotenuse is 55, the adjacent side is 0.8 × 55 = 44. The result must be shorter than the hypotenuse, which gives you a quick check on every answer.

At what age do students learn trigonometry?

In many school systems, students meet sine, cosine and tangent in right triangles at about 14 to 16. Radians and the inverse functions such as arcsin usually follow a year or two later. The levels here are set for ages 15 to 16, and they start with triangle ratios before moving on to angles, radians and the identity sin²θ + cos²θ = 1.

Why does my calculator give the wrong answer for sin 30?

Your calculator is most likely set to radians instead of degrees. In radian mode, sin 30 means the sine of 30 radians, almost five full turns, and it returns about −0.99. Switch to degree mode, usually shown as DEG or D on the display, and sin 30 gives 0.5. In this game, every angle you type is in degrees.

Is this sine cosine game free for a class?

Yes, all five levels are free for a class: they open from a link on any laptop or tablet, and no student signs up or downloads anything. As an opener for trig practice, draw right triangles with sides 3-4-5, 6-8-10 and 9-12-15. Students find the sine of the smallest angle in each, get 0.6 three times, and see that only the angle matters. The schools plan charges per student and shows the teacher who is getting faster.

Math practice · reimagined

Trigonometry Game

This trig game builds its right triangles from whole sides like 7, 24 and 25, so every sine, cosine or tangent ends as a short decimal such as 0.28. Read the triangle faster than your rival, and your fighter swings while they are still dividing.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is trigonometry?

Trigonometry is the math of how a triangle’s angles and side lengths depend on each other. In a right triangle, one sharp angle fixes the ratio between any two sides, whatever the triangle’s size. Lean a short ladder and a long one against a wall at the same angle, and each reaches up the same fraction of its length.

Every level of Trigonometry

Example: cos θ when adj is 7 and hyp is 25

The screen shows cos θ: adj 7, hyp 25, so 7 is the side next to the angle θ and 25 is the hypotenuse, the longest side.
Cosine divides the adjacent side by the hypotenuse, so cos θ = 7/25.
Multiply the top and the bottom by 4 to get hundredths: 7/25 = 28/100 = 0.28.
The answer is 0.28, and you type it with the decimal point key.

Learn it step by step

Trigonometry connects the ANGLES of a right triangle to the lengths of its sides. The three famous ratios are remembered as SOH-CAH-TOA.

SOH: sin = Opposite ÷ Hypotenuse. CAH: cos = Adjacent ÷ Hypotenuse. TOA: tan = Opposite ÷ Adjacent. (The hypotenuse is the longest side, opposite the right angle; opposite/adjacent are which leg sits across from or next to your angle.) For the 3-4-5 triangle with the angle opposite the 3-side: sin = 3 ÷ 5 = 0.6.

Pick the right ratio, divide the two sides, and type the decimal.

Tips and common mistakes

The two sharp angles of a right triangle swap labels: the side opposite one is adjacent to the other. So one angle’s sine is always the other angle’s cosine.
A common mistake is dividing upside down, 25/7 instead of 7/25. The hypotenuse is the longest side, so a sine or cosine above 1 means the fraction is flipped.
On a sunny day, stand a 1 metre stick upright and measure its shadow. Height divided by shadow is the tangent of the sun’s angle above the ground.

How to play

The first level shows two sides and a ratio to find, such as tan θ: opp 3, adj 4, and you type 0.75 with the decimal point. Later levels ask for a side, an angle like arcsin 0.5 = ? °, or π/3 rad in degrees. Only the first right answer hits, and Garlic hits hardest when you answer early.

— Quick answers —

Trigonometry: the short FAQ

What does arcsin mean?: Arcsin gives the angle whose sine is a given number, so it runs sine backwards. Since sin 30° = 0.5, arcsin 0.5 = 30°. Calculators often label it sin⁻¹, but that small −1 is not an exponent: sin⁻¹ 0.5 is an angle, not 1 divided by sin 0.5. Arccos and arctan undo cosine and tangent in the same way.
How do you find a missing side with sine or cosine?: Multiply the hypotenuse by the ratio that belongs to the side you want. Sine goes with the opposite side, and cosine with the adjacent one. If cos θ = 0.8 and the hypotenuse is 55, the adjacent side is 0.8 × 55 = 44. The result must be shorter than the hypotenuse, which gives you a quick check on every answer.
At what age do students learn trigonometry?: In many school systems, students meet sine, cosine and tangent in right triangles at about 14 to 16. Radians and the inverse functions such as arcsin usually follow a year or two later. The levels here are set for ages 15 to 16, and they start with triangle ratios before moving on to angles, radians and the identity sin²θ + cos²θ = 1.
Why does my calculator give the wrong answer for sin 30?: Your calculator is most likely set to radians instead of degrees. In radian mode, sin 30 means the sine of 30 radians, almost five full turns, and it returns about −0.99. Switch to degree mode, usually shown as DEG or D on the display, and sin 30 gives 0.5. In this game, every angle you type is in degrees.
Is this sine cosine game free for a class?: Yes, all five levels are free for a class: they open from a link on any laptop or tablet, and no student signs up or downloads anything. As an opener for trig practice, draw right triangles with sides 3-4-5, 6-8-10 and 9-12-15. Students find the sine of the smallest angle in each, get 0.6 three times, and see that only the angle matters. The schools plan charges per student and shows the teacher who is getting faster.

More math to master

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