Name the sides from the angle you are working with, and every trig question turns into a fraction.
What is the value of x?
sin(x°) = cos(58°)
where 0 < x < 90.
Sine and cosine swap at complementary angles: sin(x°) = cos(90° − x°). So cos(58°) = sin(32°), and x = 32. D says 58 — reading sin and cos as interchangeable at the same angle, which is only true at 45°. A halves 58, a guess when the identity is half-remembered. C subtracts from 100 instead of 90.
1
“Opposite” and “adjacent” are named from the angle you are working with.
The hypotenuse never moves — it is always across from the right angle. The other two labels swap the moment you switch acute angles.
Working from
opposite
adjacent
hypotenuse
angle A
8
15
17
angle B
15
8
17
The gold side faces the gold angle, the blue side faces the blue angle. Nothing about the
triangle changed between those two rows — only the angle you stood at. This is the
commonest error in the topic: computing sin A when the stem asked for sin B.
2
Three ratios, and the two sides you have pick one of them.
SOH
sin = opphyp
CAH
cos = adjhyp
TOA
tan = oppadj
sin A = 817
opposite A over hypotenuse
cos A = 1517
adjacent to A over hypotenuse
tan A = 815
opposite over adjacent — no hypotenuse
How to choose, every time
Tick the side you know and the side you want. Exactly one of the three ratios contains
both. If the hypotenuse is not involved at either end, it is tangent.
A ratio is a pure number: no units, and no change when the triangle is enlarged. Similar
right triangles have equal acute angles, so they share all three ratios — which is why the
SAT can hand you cos A in one triangle and ask for cos D in a similar one.
3
One angle and one side give you any other side.
A loading ramp rises at 14° to the horizontal and its sloped surface is 26 feet long. How high, in feet, is the top of the ramp above the ground?
known: hyp = 26, want: opp
the sloped surface is the hypotenuse
sin 14° = h26
SOH holds opp and hyp
h = 26 · sin 14°
× 26 on both sides
h = 26 · 0.2419 ≈ 6.29
feet
about 6.3 feet
Had the unknown been underneath — tan 14° = 6.29d — you divide instead:
d = 6.29 ÷ tan 14° ≈ 25.2. Unknown on top, multiply; unknown underneath, divide by the
ratio. Reaching for tan⁻¹ here is the usual wrong turn.
Set the calculator to degrees first
Desmos is on your screen all section, but it evaluates trig in radians until you change
it: open the wrench menu and switch the angle setting to Degrees. Left in radians,
sin(14) returns 0.9906 and the ramp becomes 25.8 feet — a rise of almost the full 26
feet of sloped surface, which would be a ramp standing nearly upright.
4
Two sides give you the angle, through the inverse.
In right triangle PQR, angle Q is a right angle, PQ = 9 and QR = 20. What is the measure of angle P, to the nearest degree?
from P: opp = QR = 20, adj = PQ = 9
QR faces P; PQ touches it
tan P = 209 = 2.2222
TOA
P = tan⁻¹(2.2222) ≈ 65.8°
the inverse undoes the ratio
P ≈ 66°
to the nearest degree
sin, cos and tan turn an angle into a ratio. sin⁻¹, cos⁻¹ and tan⁻¹ turn a ratio back into
an angle. What the question is missing decides which direction you need.
Sanity check: the opposite side, 20, is more than twice the adjacent, 9, so the angle must
be well past 45°. An answer of 24° would be tan⁻¹(920), the other acute angle —
and 24° + 66° = 90°, which is exactly what the next step is about.
5
sin(x°) = cos(90° − x°), and the triangle shows you why.
The two acute angles of a right triangle add to 90°, so each one is the other's complement — and they share the same sides under different names.
∠A + ∠B = 90°
the three angles total 180°, and C takes 90°
∠A = x°, so ∠B = (90 − x)°
they are complements
sin(x°) = ac
side a is opposite A
cos((90 − x)°) = ac
the same side a is adjacent to B
sin(x°) = cos(90° − x°)
same fraction, so same value
Nothing is being memorised. The gold side a faces A and touches B, so “opposite over
hypotenuse” at A and “adjacent over hypotenuse” at B are the same fraction, ac. Swap the
roles and cos(x°) = sin(90° − x°) follows the same way.
sin(x°) = cos(58°)
the opening problem
cos(58°) = sin(90° − 58°) = sin(32°)
rewrite the cosine as a sine
x = 32
and 32 + 58 = 90 ✓
x = 32
The 58 trap
Choice D in the opening problem was 58 — what you get by treating sin and cos as the same
function. They agree at exactly one acute angle, 45°, where both equal √22. Everywhere
else it is sine and cosine of complementary angles that match, never of the same
angle. Whenever a stem puts sin and cos on opposite sides of an equals sign, check
whether the two angles add to 90.
The identity also arrives dressed as a triangle: “In right triangle RST, angle T is a right
angle and sin R = 0.6. What is cos S?” R and S are the two acute angles, so they are
complementary and cos S = sin R = 0.6. You never find the angle itself.
6
One ratio plus one length rebuilds the whole triangle.
In right triangle XYZ, angle Z is a right angle and tan X = 1235. If XZ = 70, what is the perimeter of triangle XYZ?
tan X = YZXZ
YZ faces X, XZ touches it
YZ70 = 1235
70 is twice 35, so the triangle is the 12-35-37 doubled
YZ = 70 × 1235 = 24
the missing leg
XY² = 24² + 70² = 576 + 4900 = 5476
Pythagoras for the hypotenuse
XY = √5476 = 74
and 74 = 2 × 37 ✓
perimeter = 24 + 70 + 74 = 168
add all three
perimeter = 168
A tangent written as a fraction gives the shape of the triangle, not its size:
tan X = 1235 means legs in the ratio 12 : 35, which could be 12 and 35, or 24 and 70.
One real length fixes which. Spotting a triple — 3-4-5, 5-12-13, 8-15-17, 7-24-25 or
12-35-37 — saves the square root, but Pythagoras always works.
✓
Check: did you answer for the right angle?
the anglewhich vertex did the stem name?opp and adj swap between them
the functionsin, cos or tan?tan never uses the hypotenuse
size checksin and cos are always < 1a leg cannot beat the hypotenuse
complementsin(x°) = cos(90° − x°)sin 32° = cos 58° ✓
Remember the flow
Stand at the angle · Name the three sides · Pick the ratio · Solve
If the stem shows a sine and a cosine, look first at whether the angles add to 90.
Now you try
Practice problem
What is the value of cos(90° − D)?
In right triangle ABC, angle C is a right angle and
sin A = 513.
Triangle DEF is similar to triangle ABC, where A corresponds to D, B corresponds to E, and
C corresponds to F. What is the value of cos(90° − D)?
Two moves. Corresponding angles of similar triangles are equal, so ∠D = ∠A. And cos(90° − D) = sin D by the complementary-angle identity. So cos(90° − D) = sin A = 513. C is cos A — the legs are 5 and 12 with hypotenuse 13, so 1213 is the right triangle but the wrong function. B is tan A, 512.D flips the fraction, putting the hypotenuse on top; a sine can never exceed 1.
Hint: you are not being asked to find a length or an angle. Deal with the “similar” first —
it tells you ∠D equals ∠A — then read “cos of 90 minus” as an instruction to switch to sine
of the angle itself.
