12.1 Right-Triangle Trigonometry and Special Triangles

12.1 Right-Triangle Trigonometry and Special Triangles

College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights trigonometry at 5–15% of the 20-item computer-adaptive test — typically 1–3 items. Table 11 names the full cluster: solving trigonometric equations; right-triangle trig including special triangles; equivalent trigonometric functions; graphing trig relationships; arc length and radian measures; and the law of sines and the law of cosines. This section is the Skills Insight 250–262 core: simple trigonometric ratios in a right triangle. Band 276–300 extends those same ratios onto the unit circle in Unit Circle, Radians, and Arc Length. Work FREE mixed AAF items at /practice/accuplacer-advanced-algebra. Official category names live on College Board’s What’s on the Tests page.

SOH-CAH-TOA in a right triangle

Label a right triangle ABC with the right angle at C. Acute angle A has three sides relative to A:

• Opposite is the side that does not touch A (here, side BC).
• Adjacent is the leg that does touch A but is not the hypotenuse (here, side AC).
• Hypotenuse is the side opposite the right angle (here, side AB). It is always the longest side.

The three primary ratios are:

sin A = opposite / hypotenuse

cos A = adjacent / hypotenuse

tan A = opposite / adjacent

The mnemonic SOH-CAH-TOA is the whole skill: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. AAF will not grade you on reciting the letters; it will grade you on picking the correct two sides for the named acute angle. Switch the named angle from A to B and opposite and adjacent trade places. The hypotenuse never moves.

Reciprocal identities

The three reciprocal functions flip the primary ratios:

csc A = 1 / sin A = hypotenuse / opposite

sec A = 1 / cos A = hypotenuse / adjacent

cot A = 1 / tan A = adjacent / opposite

If you already have sin A, csc A is its reciprocal — do not rebuild it from a different pair of sides. Two more compact identities sit on top of the same triangle: tan A = sin A / cos A and cot A = cos A / sin A. Reciprocal does not mean complementary. csc A is the flip of sin A; it is not cos B.

Worked example: sin A = 5/13, then every other ratio

Right triangle ABC has a right angle at C, and sin A = 5/13. Find cos A, tan A, csc A, and sin B.

Because sin A = opposite / hypotenuse = 5/13, the opposite side may be taken as 5 and the hypotenuse as 13 (any positive multiple of that pair gives the same ratio). The missing adjacent side comes from the Pythagorean theorem:

adjacent^2 + 5^2 = 13^2

adjacent^2 + 25 = 169

adjacent^2 = 144

adjacent = 12

(A length is positive, so discard −12.) This is the 5-12-13 triple. Now every ratio is a fraction of those three numbers:

• cos A = 12/13
• tan A = 5/12
• csc A = 13/5
• sec A = 13/12
• cot A = 12/5

Train on this triple. Do not drill a published sample that starts from a cosine of five-eighths; AAF will reuse the method, not that particular pair of sides.

Complementary angles: sin A = cos B

In right triangle ABC with right angle C, the two acute angles are complementary: A + B = 90°. The side opposite A is adjacent to B, and the side opposite B is adjacent to A. Therefore the cofunction identities are not extra formulas — they are the same two legs, read from the other angle:

sin A = cos B

cos A = sin B

tan A = cot B

From the 5-12-13 example, sin A = 5/13, so cos B = 5/13 as well — not 12/13. The trap is treating complementary as “the same angle.” Complementary means the angles add to 90°; the cofunction identity swaps sine with cosine. The same identity in special-angle clothing is sin 30° = cos 60° = 1/2 and sin 45° = cos 45° = √2/2.

sin B uses the side opposite B, which is the adjacent to A, so sin B = 12/13. That matches cos A = 12/13, as the cofunction rule promised.

Finding the third side first

AAF often gives two sides and asks for a trig ratio that needs the third. Always sketch, label opposite/adjacent/hypotenuse relative to the named acute angle, then apply Pythagoras if a side is missing. Pythagoras for right triangle ABC with right angle C is a^2 + b^2 = c^2, the same identity taught with the distance formula in Distance Formula and Pythagorean Theorem.

Worked: Legs 8 and 15, hypotenuse unknown, angle A opposite the side of length 8. Then hypotenuse = √(8^2 + 15^2) = √(64 + 225) = √289 = 17. So sin A = 8/17, cos A = 15/17, tan A = 8/15. If the stem had instead said angle A is adjacent to the 8, you would swap: sin A = 15/17. The number 17 does not tell you which ratio — the words opposite versus adjacent do.

Keep 7-24-25 and 20-21-29 as extra triples you can finish quickly, but you do not need a catalog. You need to put the hypotenuse in the c slot and not in a leg slot. A stem that already gives a hypotenuse of 25 and a leg of 7 is one subtraction away from 24: adjacent^2 = 25^2 − 7^2 = 625 − 49 = 576 = 24^2.

Special triangle: 45-45-90

An isosceles right triangle has acute angles 45° and 45°. If each leg is x, the hypotenuse is x√2. The two legs are equal, so tan 45° = x/x = 1 at every scale.

Worked: Each leg is 7. Hypotenuse = 7√2. Then sin 45° = 7 / (7√2) = 1/√2 = √2/2 after rationalizing. If the hypotenuse is 10√2, each leg is 10, not 10√2. Divide by √2: leg = hyp / √2 = (10√2)/√2 = 10.

Worked from a hypotenuse without a radical already showing: Hypotenuse 18. Then x√2 = 18, so x = 18/√2 = 9√2. Each leg is 9√2, and sin 45° is still √2/2 because the scale cancels.

Trap: treating a 45-45-90 like a 30-60-90 and putting √3 on a side. There is no √3 in a 45-45-90 triangle. A second trap: writing the hypotenuse as 2x because “the hypotenuse is twice a leg.” That doubling rule belongs only to 30-60-90.

Special triangle: 30-60-90

A 30-60-90 triangle has sides in the ratio x : x√3 : 2x.

• Side opposite 30° is the shortest leg, x.
• Side opposite 60° is the longer leg, x√3.
• Hypotenuse is opposite 90°, 2x.

Worked from the short leg: The side opposite 30° is 5. Then x = 5, hypotenuse 2x = 10, and the side opposite 60° is 5√3. So sin 30° = 5/10 = 1/2, cos 30° = (5√3)/10 = √3/2, sin 60° = (5√3)/10 = √3/2, cos 60° = 5/10 = 1/2.

Worked from the hypotenuse: Hypotenuse 14. Then 2x = 14, so x = 7. Opposite 30° is 7; opposite 60° is 7√3.

Worked from the long leg: Opposite 60° is 9√3. Then x√3 = 9√3, so x = 9. Opposite 30° is 9; hypotenuse is 18. If a stem instead gives the long leg as 9 with no radical, then x√3 = 9, so x = 9/√3 = 3√3, hypotenuse 6√3, and opposite 30° is 3√3.

Special-angle value table (exact)

Memorize the exact values. AAF will mix them with algebra — 2 sin 30° + cos 60° is 2(1/2) + 1/2 = 3/2, not a calculator decimal. tan 30° · tan 60° = (√3/3) · √3 = 1, which is also tan 45°.

Sine increases through 0, 1/2, √2/2, √3/2, 1 while cosine decreases through the same five numbers in reverse. That is the complementary pattern again: sin 30° = cos 60°. Tangent is sine over cosine, which is why tan 60° = (√3/2) / (1/2) = √3 and why tan 90° is undefined (division by cosine zero).

AAF traps for right-triangle trig

• Wrong pair of sides. tan never uses the hypotenuse. If you divide a leg by the hypotenuse you have sine or cosine, not tangent.
• Wrong reference angle. Opposite and adjacent switch when you move from angle A to angle B. Compute sin A and sin B separately after you label.
• Leaving Pythagoras unfinished. A stem that gives two legs and asks for sin A is not a two-side ratio until you find the hypotenuse.
• Exact versus decimal. sin 45° = √2/2, not 0.7. If every option is exact, do not pick a rounded neighbor.
• Reciprocal mix-up. csc A = 13/5 is the flip of sin A = 5/13. 13/5 is not cos A.
• 30-60-90 assignment. The √3 always sits opposite 60°, never opposite 30°. Opposite 30° is half the hypotenuse.
• Complementary mix-up. sin A = cos B, not sin A = cos A (unless A = 45°).

Handheld calculators are not allowed except with an approved accommodation (College Board calculator policy). Some AAF items show an on-screen calculator icon at the top-right corner; special-angle items are written so the exact value is faster than a decimal. QAS and Arithmetic do not carry this 30-60-90 / cofunction load — those lighter placement tests live at /study-guides/accuplacer-qas and /study-guides/accuplacer-arithmetic. AAF is the STEM placement path that puts right-triangle trig in the 250–262 band and then asks you to keep the same ratios on the unit circle at 276–300.

Mini mixed set (work these before the quizzes)

1. Right triangle, right angle C, a = 9, c = 15. Then b = 12 (3-4-5 scaled by 3). If a is opposite A, sin A = 9/15 = 3/5, cos A = 12/15 = 4/5, tan A = 9/12 = 3/4, and csc A = 5/3.
2. 45-45-90 with hypotenuse 6√2. Legs = 6. tan 45° = 1 regardless of the scale, and sec 45° = √2.
3. 30-60-90 with longer leg 10√3. Then x√3 = 10√3, x = 10, hypotenuse 20, sin 60° = (10√3)/20 = √3/2, tan 30° = 10 / (10√3) = √3/3.
4. Right triangle ABC, right angle C, sin A = 20/29. Adjacent = √(29^2 − 20^2) = √(841 − 400) = √441 = 21. Then cos A = 21/29 and cos B = 20/29 by the cofunction rule.

That is the whole 250–262 skill: name the sides, finish Pythagoras, quote an exact special-angle value when the triangle is 30-60-90 or 45-45-90, and remember that complementary acute angles swap sine with cosine.