AP Physics Trig Table: sin, cos, tan Common Angles

The Table Exactly as the Booklet Prints It

The College Board calls this box VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES. It sits on the Table of Information page of the reference booklet, next to the prefixes and the unit symbols, and the same seven columns appear in all four Course and Exam Descriptions effective Fall 2024:

Seven angles, three functions, twenty one values. That is the whole table. There is no cotangent row, no secant row, no cosecant row, and no inverse-function column.

Where it is printed:

• AP Physics 1: Algebra-Based, appendix page 211
• AP Physics 2: Algebra-Based, appendix page 218
• AP Physics C: Mechanics, appendix page 204
• AP Physics C: Electricity and Magnetism, appendix page 178

The four printings are character for character identical, so a student moving from Physics 1 to Physics C carries exactly the same trig table into the room.

Notice What the Booklet Chose Not to Round

Every entry above is an exact fraction or an exact surd. The booklet prints $\sqrt{2}/2$, not 0.707. It prints $\sqrt{3}/3$ for $\tan 30°$, not 0.577, and not the equivalent-looking $1/\sqrt{3}$: the rationalised form is what appears.

That is a deliberate choice and it tells you something about the exam. Free-response answers are graded on the physics, so an answer left as $mg\sqrt{3}/2$ is complete. You do not gain credit for converting it to 8.5 N, and you can lose the thread of the algebra doing so.

For reference, the decimal equivalents of the four surds are:

The multiple-choice section is where you will want those, because the distractors are usually far enough apart that one decimal place settles it.

Radians, Which the Booklet Does Not Give You

The printed table has a degrees row and nothing else. No AP Physics booklet prints a radian column for these angles. The table below adds one, and it is our addition, not a transcription.

The awkward two rows are the tell. Thirty, forty five, sixty and ninety degrees all land on tidy fractions of $\pi$, while 37° and 53° do not, because they were never chosen for their radian value. The next section says what they were chosen for.

Radians matter in one specific place: the arc-length relation $s = r\theta$ printed in the geometry box, and every rotational formula built on it, is only true when $\theta$ is in radians. Feed it degrees and your answer is wrong by a factor of $180/\pi$. See the geometry table for where that relation sits on the sheet.

Why 37 and 53 Degrees Are on a Physics Table

They are the 3-4-5 right triangle. A triangle with legs 3 and 4 and hypotenuse 5 has a smaller acute angle of $\arctan(3/4) = 36.87°$ and a larger one of $53.13°$. The booklet rounds those to 37° and 53° and prints the exact 3-4-5 ratios against them.

So the 37° and 53° columns are not approximations of a hard number, they are exact values of a slightly different angle. The size of the gap:

The worst of those is $\tan 53°$, where the printed 4/3 runs about 0.5 percent high. Nothing on an AP exam is graded to a tolerance that tight, and the exam writers put 37° and 53° in problems precisely so the arithmetic stays clean.

The practical consequence: when a problem hands you 37° or 53°, use 3/5, 4/5, 3/4 and 4/3. You will finish faster than the student reaching for a calculator, and you will get the answer the rubric expects.

Which Column You Actually Need, By Problem Type

The table is short enough to read in one glance, so the useful question is not what it says but which entry a given problem wants.

• Inclined planes. The weight splits into $mg\sin\theta$ down the slope and $mg\cos\theta$ into the surface, where $\theta$ is measured from the horizontal. At 30° the split is $mg/2$ and $mg\sqrt{3}/2$, which is why 30° inclines make such tidy problems.
• Launching a projectile. $v_{x} = v_0\cos\theta$ and $v_{y0} = v_0\sin\theta$. A 53° launch gives components in a 3 to 4 ratio, so a 25 m/s launch is exactly 15 and 20.
• Tension in two ropes. Both the sine and the cosine of each rope angle appear, one in the vertical equation and one in the horizontal.
• Torque. The sheet prints $\tau = r_{\perp}F = rF\sin\theta$, so torque problems always want the sine row.
• Work at an angle. The sheet prints $W = F_{\parallel}d = Fd\cos\theta$, so work problems always want the cosine row. A force at 90° to the displacement does zero work, which is the $\cos 90° = 0$ entry doing its job.
• Magnetic force. $F_B = qvB\sin\theta$ and $F_B = I\ell B\sin\theta$ are both sine, and both vanish at $\theta = 0$, which is the $\sin 0° = 0$ entry.

The pattern worth memorising is that dot-product quantities carry a cosine and cross-product quantities carry a sine. For the calculus-based statement of that, see the vector operations table.

Rebuilding the Table If You Blank

You will have the booklet in the exam, so this is insurance rather than a requirement. Two triangles generate every entry.

The 30-60-90 triangle. Take an equilateral triangle of side 2 and drop a perpendicular. You get a right triangle with hypotenuse 2, short leg 1 opposite the 30° angle, and long leg $\sqrt{3}$ opposite the 60° angle. Read the ratios straight off: $\sin 30° = 1/2$, $\cos 30° = \sqrt{3}/2$, $\tan 30° = 1/\sqrt{3} = \sqrt{3}/3$. Swap the roles of the legs for 60°.

The 45-45-90 triangle. Legs of 1 and 1, hypotenuse $\sqrt{2}$. So $\sin 45° = \cos 45° = 1/\sqrt{2} = \sqrt{2}/2$ and $\tan 45° = 1$.

The 3-4-5 triangle gives the 37° and 53° columns, as above.

Two structural facts fill in the rest. First, sine and cosine are mirror images across the table: $\cos\theta = \sin(90° - \theta)$, so the cosine row is the sine row read backwards. Check it against the printed table and you will see the reversal holds in all seven columns. Second, $\tan\theta = \sin\theta / \cos\theta$, which is one of the four identities printed for Physics C and is listed on the identities page.

The Infinity Entry, and the Other Edges

The booklet prints $\infty$ for $\tan 90°$. Strictly the tangent is undefined there, because $\cos 90° = 0$ and you cannot divide by zero. The symbol records the behaviour rather than a value: as $\theta$ climbs toward 90°, $\tan\theta$ grows without bound.

In physics that entry usually flags a setup that has broken down. A rope pulled perfectly horizontal to support a hanging weight needs infinite tension; a banked curve at 90° needs infinite speed. If a tangent in your working is heading for infinity, check the geometry before you check the algebra.

The other two edge columns are worth reading deliberately, because they are the ones students misremember under pressure:

• At $0°$: $\sin = 0$, $\cos = 1$, $\tan = 0$. A force along the direction of motion does maximum work and exerts zero torque.
• At $90°$: $\sin = 1$, $\cos = 0$, $\tan = \infty$. A force perpendicular to the motion does zero work and exerts maximum torque.

Those two lines are the whole content of the sine-versus-cosine confusion, resolved.

What This Table Is Not

The booklet gives you seven angles. It does not give you a way to handle the other 353, and that is intentional: AP problems are written around the values in the table.

Specifically, none of the following is printed anywhere in any of the four booklets, so do not go looking:

• Sine, cosine or tangent of any angle other than the seven listed.
• Inverse trig values, so no table of $\arcsin$ or $\arctan$.
• Values beyond 90°, so no second, third or fourth quadrant entries and no sign conventions for them.
• The law of sines or the law of cosines.
• The small-angle approximation $\sin\theta \approx \theta$, despite it being assumed in the pendulum treatment. The AP Physics 2 conventions box does state that the small-angle approximation is valid for single- and double-slit diffraction, but that is a stated exam convention rather than a printed formula.

For angles outside the table, the exam expects a calculator. All four CEDs state the same policy: a four-function, scientific or graphing calculator is allowed on both sections of the exam. What the table saves you is not capability, it is time.