AP Physics C Identities: The Four Printed Lines

The Four Lines, as Printed

The box is titled IDENTITIES, not trigonometric identities, and the title is accurate: one of its four entries is about logarithms. It is the rightmost of three boxes on the last appendix page, sharing a row with VECTORS and CALCULUS.

Four lines. That is the entire box, and the count is worth stating plainly because a short list invites the assumption that more is hiding elsewhere on the page. It is not.

Where it is printed:

• AP Physics C: Mechanics, appendix page 206
• AP Physics C: Electricity and Magnetism, appendix page 181

The two printings are identical. Neither algebra-based booklet prints an identities box. AP Physics 1 puts its whole reference on appendix pages 211 and 212, and AP Physics 2 on 218 through 221, and no identity appears on any of them. The only trigonometry an algebra-based candidate gets is the geometry box and the table of trig values, both of which are shared with Physics C.

The slug for this page says trigonometric because that is what students search for. The booklet's own heading is broader, and the logarithm entry is the reason.

The Logarithm Rule Is Three Rules in One Line

$\log(a \cdot b^x) = \log a + x \log b$ looks like an odd, over-specific thing to print. It is actually the only logarithm line the booklet needs, because the three rules you were taught separately all fall out of it.

• Set $x = 1$: $\log(a \cdot b) = \log a + \log b$. The product rule.
• Set $x = -1$: $\log(a/b) = \log a - \log b$. The quotient rule.
• Set $a = 1$: $\log(b^x) = x\log b$, since $\log 1 = 0$. The power rule.

The booklet does not state a base, and it does not need to: the identity is true for any base, including base 10 and the natural logarithm. Choosing the base is your decision, and for physics the natural logarithm is usually the right one, for the reason in the next paragraph.

What it is printed for is linearisation. AP Physics C is full of quantities that decay or grow exponentially: charge on a discharging capacitor, current in an LR circuit, the speed of an object under a resistive force proportional to velocity. Every one of those has the form $y = y_0 e^{-t/\tau}$, and every one of them becomes a straight line when you take a logarithm:

Plot $\ln y$ against $t$ and the slope is $-1/\tau$ and the intercept is $\ln y_0$. That is precisely the manipulation the AP experimental-design questions ask for when they say to choose axes that produce a linear graph. Use base 10 instead and the slope picks up a factor of $\log e = 0.4343$, which is a real answer but an easy one to forget to include.

The C: E&M sheet prints $\tau = R_{eq}C_{eq}$ for an RC circuit and $\tau = L/R_{eq}$ for an LR circuit, so once you have the slope you have the time constant, and from the time constant you have the capacitance or the inductance. That chain is why one logarithm identity earns a place on a physics sheet.

The Pythagorean Identity, and Where It Hides

$\sin^2\theta + \cos^2\theta = 1$ is the right-triangle relation $a^2 + b^2 = c^2$ divided through by $c^2$. It is printed in the identities box, and its parent is printed in the geometry box on the same page. Two statements of the same fact, two boxes apart.

In AP Physics C it does three jobs:

Recovering a magnitude from components. If $A_x = A\cos\theta$ and $A_y = A\sin\theta$, then $A_x^2 + A_y^2 = A^2(\cos^2\theta + \sin^2\theta) = A^2$. That is the derivation of $A = \sqrt{A_x^2 + A_y^2}$, which is a formula the booklet does not print anywhere. The identities box gives you the ingredient rather than the result.

Proving energy is conserved in simple harmonic motion. The sheet prints $x = x_{max}\cos(\omega t + \phi)$. Differentiating gives $v = -\omega x_{max}\sin(\omega t + \phi)$. Then

and with $\omega^2 = k/m$ the two coefficients match, so the identity collapses the sum to $\frac{1}{2}kx_{max}^2$, a constant. The total energy of an oscillator is constant precisely because of this identity.

Relating the two components of a magnetic or electric field at an angle. Whenever a problem gives you one component and the total, this identity gives you the other without a second trig lookup.

One warning about notation. $\sin^2\theta$ means $(\sin\theta)^2$, not $\sin(\sin\theta)$ and not $\sin(\theta^2)$. The booklet prints the exponent between the function name and the argument, which is standard and still trips people writing it into a calculator.

The Double-Angle Formula and the Range Equation

$\sin(2\theta) = 2\sin\theta\cos\theta$ has one dominant use on an AP exam, and it is projectile range.

The range equation $R = \dfrac{v_0^2 \sin(2\theta)}{g}$ is not printed on any of the four sheets. What is printed is the pair of kinematic equations and, in the identities box, the double-angle formula that turns the honest two-step calculation into that one-liner. Derive it once:

A projectile launched at speed $v_0$ and angle $\theta$ over level ground has $v_x = v_0\cos\theta$ and $v_{y0} = v_0\sin\theta$. Flight time is $t = 2v_{y0}/g$. Range is $v_x t$:

The identity did the last step and nothing else. Two consequences you can read straight off the result:

• Maximum range is at 45°, because $\sin(2\theta)$ peaks when $2\theta = 90°$.
• Complementary launch angles give the same range. A 30° launch and a 60° launch land in the same place, because $\sin 60° = \sin 120°$. The 60° shot spends longer in the air and travels slower horizontally, and the two effects cancel exactly.

The booklet prints the sine double-angle formula and not the cosine one. There is no $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ anywhere, and no $1 - \cos(2\theta) = 2\sin^2\theta$, which is the identity you would want for averaging $\sin^2$ over a cycle in an AC or wave problem. If you need it, derive it from the two identities you do have, or memorise it.

Tangent as a Ratio, and Why That Line Is Not Redundant

$\dfrac{\sin\theta}{\cos\theta} = \tan\theta$ looks like the least useful line in the box. It is printed because it is the step that converts a two-equation problem into a one-equation problem, and that conversion appears constantly.

The pattern: you write Newton's second law twice, once along each axis, and both equations contain the same unknown magnitude. Divide one by the other and the magnitude cancels, leaving a tangent.

• Angle of repose. At the point of slipping, $mg\sin\theta = \mu_s mg\cos\theta$, so $\tan\theta = \mu_s$. The mass and $g$ both vanish, which is why the angle at which a block starts to slide tells you the coefficient of friction and nothing else.
• Banked curve with no friction. The normal force satisfies $N\sin\theta = mv^2/r$ horizontally and $N\cos\theta = mg$ vertically. Divide: $\tan\theta = v^2/(rg)$. The normal force and the mass both cancel.
• Direction of a resultant. Given $A_x$ and $A_y$, the angle from the $x$ axis is the one whose tangent is $A_y/A_x$. Note that the booklet gives you the identity, not the inverse function, and a calculator's inverse tangent cannot tell a second-quadrant angle from a fourth-quadrant one. Sketch the vector before you trust the number.
• Charged pendulum in a uniform field. A hanging charge deflected by a horizontal electric field settles where $\tan\theta = qE/(mg)$, by exactly the same divide-the-two-equations move.

The general lesson is worth more than the line itself: when two force equations share an unknown, dividing beats substituting.

What the Booklet Leaves Out

This list matters more than usual, because the identities box is short enough that students assume the rest is somewhere else on the page. It is not. None of the following appears in either AP Physics C booklet:

• The cosine double-angle formula, in any of its three forms.
• Sum and difference formulas, so no $\sin(A \pm B)$ and no $\cos(A \pm B)$.
• Half-angle formulas.
• $1 + \tan^2\theta = \sec^2\theta$, or any identity involving secant, cosecant or cotangent. Those three functions do not appear anywhere in any AP Physics booklet.
• The law of sines and the law of cosines. AP vector addition is done by components.
• Even and odd relations, so no $\sin(-\theta) = -\sin\theta$ and no $\cos(-\theta) = \cos\theta$.
• The small-angle approximation $\sin\theta \approx \theta$, despite the simple-pendulum period being on the sheet and depending on it. AP Physics 2 does state as an exam convention that the small-angle approximation is valid for single- and double-slit diffraction, but that is a convention bullet, not a printed identity, and it is in a different course's booklet.
• Euler's formula or any complex-exponential representation.
• Product-to-sum formulas, which is what a full treatment of beats would need.
• Change of base for logarithms, so no $\log_b x = \ln x / \ln b$.

One of those omissions is a genuine gap rather than a simplification. Averaging $\sin^2$ or $\cos^2$ over a full cycle gives $\frac{1}{2}$, which is where the root-mean-square factor of $\sqrt{2}$ comes from, and the clean route to it runs through the cosine double-angle formula the booklet does not print. If your course covers it, that identity is worth carrying in your head.

A note on how to check any claim like this. Transcriptions of the AP equation sheet, including several widely circulated ones, cover the physics equations and the constants and quietly omit the geometry, trigonometry, vector, calculus and identity boxes. A formula missing from such a transcription may still be printed in the booklet. Everything on this page was read from a rendered image of appendix page 206 of the C: Mechanics CED and appendix page 181 of the C: E&M CED.

Physics 1 and Physics 2 Candidates: What You Get Instead

If you are sitting an algebra-based exam, the identities box is not in your booklet, and you should not plan around it. What you do get, shared word for word with Physics C:

• The GEOMETRY AND TRIGONOMETRY box, which includes $\sin\theta = a/c$, $\cos\theta = b/c$, $\tan\theta = a/b$ and $a^2 + b^2 = c^2$. Between them those four give you the Pythagorean identity and the tangent ratio, just stated in terms of triangle sides rather than functions.
• The VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES table, covering 0, 30, 37, 45, 53, 60 and 90 degrees.

What you genuinely do not have is the double-angle formula. Projectile-range questions in AP Physics 1 are therefore built to be answered by components and flight time rather than by the range equation, and that two-step route is the one the rubrics are written around. It is also the route that survives a question set on a slope or from a cliff, where the range equation does not apply at all.

The practical difference between the courses here is smaller than the boxes suggest. See the geometry table and the trig values table for the two boxes every course does share.