In a right triangle, each acute angle can serve as a reference angle, and the three trigonometric ratios — sine, cosine, and tangent — compare the sides of the triangle from that angle’s point of view. This lesson names the sides from the reference angle’s point of view, writes the three ratios with the SohCahToa mnemonic, evaluates the ratios from side lengths, and uses the ratios to find missing sides. Trig values are rounded to 4 decimal places (ten-thousandths) while working, and final side measurements are rounded to the nearest tenth.
Trigonometry in a right triangle starts with three ratios — sine, cosine, and tangent — each comparing two sides relative to a chosen reference angle. This lesson names the sides (hypotenuse, opposite, adjacent) from the reference angle's point of view, writes the ratios with the SohCahToa mnemonic, evaluates ratios from side lengths, and uses the ratios to find missing sides. Before any calculation, confirm the calculator is in degree mode: sin 30° must return 0.5.
Calculator check: Always confirm the setting of the calculator. Test: sin 30° = 0.5. If you get anything other than 0.5, your calculator is not set to degrees.
What you are looking at: A right triangle with vertices A, B, and C. The right angle is at C. The pale gold wedge marks the reference angle, ∠CAB. Every side has a role relative to that angle: the side directly across from it is the opposite side, the side next to it that is not the hypotenuse is the adjacent side, and the side across from the right angle is the hypotenuse.
What to do: Use the buttons below the diagram to highlight each side and read its role in the readout. Then switch the reference angle to the other acute angle and watch: the opposite and adjacent sides trade places, and the hypotenuse does not change. After exploring, try the Name This Side check — each question names a role, and you click the side that plays that role for the reference angle shown.
Switch the reference angle to the other acute angle. Watch the opposite and adjacent sides trade places; the hypotenuse stays the hypotenuse.
Highlight one side at a time. The readout below names that side’s role relative to the reference angle, in a complete sentence.
What to do: Each question shows a reference angle in the diagram and asks for one side’s role. Click the side that plays that role. The side you choose is drawn in the diagram, whether the choice is right or wrong, so you can see what the choice means. A first miss gives a hint and another try; a second miss names the correct side and explains it. The six questions take turns in order.
What you are looking at: A right triangle with the right angle at C and the reference angle at A, marked ∠CAB. SohCahToa states the three ratios in full, for the reference angle ∠CAB: sine (sin ∠CAB) = oppositehypotenuse, cosine (cos ∠CAB) = adjacenthypotenuse, and tangent (tan ∠CAB) = oppositeadjacent. The readout computes all three ratios live from the side lengths in the diagram, first as fractions of the side lengths and then as values rounded to 4 decimal places. The values are computed from the full-precision lengths and rounded only at the end.
What to do: Drag the orange handle on point A along the bottom to change the adjacent side, and the purple handle on point B up or down to change the opposite side — both change the reference angle. Use the Smaller and Larger buttons to change only the size of the triangle while its angles stay the same.
What to notice: Dragging a vertex changes the side lengths, and changing the reference angle changes the ratio values. Changing only the size — with the Smaller and Larger buttons — changes the side lengths but leaves the angle and all three ratio values the same. The ratios belong to the angle, not to the size of the triangle.
Change only the size of the triangle: every side grows or shrinks by the same factor, so the angles — and the three ratio values in the readout — stay the same.
What to do: Each problem gives a right triangle with one side missing. Work in four steps: (1) choose the ratio, (2) choose the equation, (3) work with the ratio’s value, rounded to 4 decimal places, and (4) report the answer, rounded to the nearest tenth when it is not exact. The pale gold wedge marks the reference angle, and the bold gold highlight marks the missing side; the missing side is labeled with its measure only after you find it. A first miss gives a hint and another try; a second miss names the correct choice and explains it. Press New Item for the next problem, in order.
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Trigonometry (Right Triangles) — Mr. J Geometry. All activities on this page run entirely in your browser.