Chapter 6 — Right Triangle Approach
Angles are far easier to measure than distances. The trigonometric functions give us the tools to convert measured angles into distances we could never reach with a tape measure — from the height of a tree to the distance to the sun. This chapter defines the trigonometric functions as functions of angles and applies them to solving both right and oblique triangles.
Chapter-level note
This chapter is the backbone of nearly every statics and dynamics problem you'll meet in mechanical engineering. Force resolution, truss analysis, and linkage geometry all reduce to "solve the triangle." Master Sections 6.2, 6.5, and 6.6 cold — you will use them every single week.
Angle Measure
Angle, Initial & Terminal Sides
An angle AOB consists of two rays \(R_1\) and \(R_2\) sharing a common vertex \(O\). Interpreting an angle as a rotation of \(R_1\) onto \(R_2\), the ray \(R_1\) is the initial side and \(R_2\) is the terminal side. A counterclockwise rotation gives a positive angle; a clockwise rotation gives a negative angle.
Radian Measure
If a circle of radius 1 is drawn with the vertex of an angle at its center, the measure of the angle in radians is the length of the arc that subtends the angle. Since the unit circle has circumference \(2\pi\), a complete revolution is \(2\pi\) rad, a straight angle is \(\pi\) rad, and a right angle is \(\pi/2\) rad.
To convert degrees → radians, multiply by \(\dfrac{\pi}{180}\). To convert radians → degrees, multiply by \(\dfrac{180}{\pi}\).
Standard Position & Coterminal Angles
An angle is in standard position if its vertex is at the origin and its initial side lies on the positive \(x\)-axis. Two angles in standard position are coterminal if their terminal sides coincide. Coterminal angles are found by adding or subtracting whole multiples of \(360^\circ\) (or \(2\pi\) rad).
In a circle of radius \(r\), an arc subtending a central angle of \(\theta\) radians has length \(s\). The formula holds only when \(\theta\) is measured in radians.
The area \(A\) of a sector of central angle \(\theta\) radians in a circle of radius \(r\). Again, \(\theta\) must be in radians.
Linear Speed & Angular Speed
For a point moving along a circle of radius \(r\) whose ray traverses \(\theta\) radians in time \(t\) (with distance \(s = r\theta\)): angular speed measures how fast the angle changes, while linear speed measures how fast distance is covered. Angular speed does not depend on the radius; linear speed does.
Section 6.1 note
Watch your mode! Every formula here (\(s=r\theta\), \(A=\tfrac12 r^2\theta\), \(v=r\omega\)) demands radians. The single most common error I see is plugging in degrees. In rotating-machinery problems, convert rpm to rad/s first: multiply rpm by \(2\pi/60\). Make that conversion a reflex.
Trigonometry of Right Triangles
The Trigonometric Ratios
For a right triangle with acute angle \(\theta\), the six trigonometric ratios are defined from the side opposite \(\theta\), the side adjacent to \(\theta\), and the hypotenuse. Because all right triangles sharing angle \(\theta\) are similar, the ratios depend only on \(\theta\), not on the size of the triangle.
| \(\theta^\circ\) | \(\theta\) rad | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) | \(\csc\theta\) | \(\sec\theta\) | \(\cot\theta\) |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 | — | 1 | — |
| 30 | \(\pi/6\) | \(\tfrac12\) | \(\tfrac{\sqrt3}{2}\) | \(\tfrac{\sqrt3}{3}\) | 2 | \(\tfrac{2\sqrt3}{3}\) | \(\sqrt3\) |
| 45 | \(\pi/4\) | \(\tfrac{\sqrt2}{2}\) | \(\tfrac{\sqrt2}{2}\) | 1 | \(\sqrt2\) | \(\sqrt2\) | 1 |
| 60 | \(\pi/3\) | \(\tfrac{\sqrt3}{2}\) | \(\tfrac12\) | \(\sqrt3\) | \(\tfrac{2\sqrt3}{3}\) | 2 | \(\tfrac{\sqrt3}{3}\) |
| 90 | \(\pi/2\) | 1 | 0 | — | 1 | — | 0 |
Solving a Right Triangle & Angles of Elevation/Depression
To solve a triangle means to determine all three sides and all three angles. If the hypotenuse \(r\) and an acute angle \(\theta\) are known, the legs are \(a = r\sin\theta\) and \(b = r\cos\theta\). The angle of elevation is measured upward from the horizontal line of sight; the angle of depression is measured downward. When the sight line follows a physical slope, it is the angle of inclination.
Section 6.2 note
"SOH-CAH-TOA" gets you started, but memorize the \(30\)–\(60\)–\(90\) and \(45\)–\(45\)–\(90\) triangles instead of the value table — you can rebuild every entry from two triangles. In the field, the elevation/depression distinction trips people up: the angle of depression from the plane equals the angle of elevation from the ground (alternate interior angles). Draw the horizontal dashed line every time.
Trigonometric Functions of Angles
Extending the Ratios to Any Angle
Place angle \(\theta\) in standard position and let \(P(x,y)\) be any point on its terminal side at distance \(r=\sqrt{x^2+y^2}\) from the origin. The trigonometric functions are then defined for any angle, not just acute ones. Their values are independent of which point \(P\) is chosen because all such triangles are similar.
| Quadrant | Positive functions | Negative functions |
|---|---|---|
| I | all | none |
| II | sin, csc | cos, sec, tan, cot |
| III | tan, cot | sin, csc, cos, sec |
| IV | cos, sec | sin, csc, tan, cot |
Mnemonic: All Students Take Calculus (Quadrants I–IV: All, Sine, Tangent, Cosine positive).
Reference Angle
The reference angle \(\bar\theta\) for an angle \(\theta\) in standard position is the acute angle formed by the terminal side of \(\theta\) and the \(x\)-axis. To evaluate a trig function of any \(\theta\): (1) find the reference angle \(\bar\theta\); (2) determine the sign from the quadrant; (3) the value equals that of the function at \(\bar\theta\), with the appropriate sign.
Reciprocal & Quotient
\[ \csc\theta=\frac{1}{\sin\theta}\quad \sec\theta=\frac{1}{\cos\theta}\quad \cot\theta=\frac{1}{\tan\theta} \] \[ \tan\theta=\frac{\sin\theta}{\cos\theta}\qquad \cot\theta=\frac{\cos\theta}{\sin\theta} \]Pythagorean
\[ \sin^2\theta+\cos^2\theta=1 \qquad \tan^2\theta+1=\sec^2\theta \qquad 1+\cot^2\theta=\csc^2\theta \]Valid for any included angle \(\theta\), acute or obtuse, since \(\sin(180^\circ-\theta)=\sin\theta\).
Section 6.3 note
The \(\tfrac12 ab\sin\theta\) area formula is gold for mechanics — it's the magnitude of a cross product \(|\vec a\times\vec b|\) in disguise, which is exactly how you compute moments (torque) of a force about a point. When you reach statics, you'll recognize this formula as the area of the parallelogram spanned by two vectors. Tie that knot now.
Inverse Trigonometric Functions and Right Triangles
Why Restrict the Domain?
The trigonometric functions are not one-to-one, so they have no inverse over their full domains. By restricting each to an interval on which it attains every value exactly once, we obtain one-to-one functions with the same range that do have inverses, written \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\) (also called arcsine, arccosine, arctangent).
| Function | Domain | Range |
|---|---|---|
| \(\sin^{-1}\) | \([-1,1]\) | \([-\pi/2,\;\pi/2]\) |
| \(\cos^{-1}\) | \([-1,1]\) | \([0,\;\pi]\) |
| \(\tan^{-1}\) | \((-\infty,\infty)\) | \((-\pi/2,\;\pi/2)\) |
Solving Basic Trig Equations on \([0^\circ,180^\circ]\)
To prepare for the Laws of Sines and Cosines (where triangle angles lie in \((0^\circ,180^\circ)\)): for sine, the calculator gives one solution in \([-\pi/2,\pi/2]\); the second is its supplement \(180^\circ-\theta\). For cosine, which is one-to-one on \([0,\pi]\), there is only one solution in that range.
Section 6.4 note
Your calculator's \(\sin^{-1}\) only ever returns one angle — and for sine that's often not the one your physical problem wants. Always sketch the triangle or the standard-position picture and ask, "could the supplement also work?" This is the seed of the ambiguous case in 6.5. Also: most calculators lack \(\sec^{-1}, \csc^{-1}, \cot^{-1}\); use \(\sec^{-1}x=\cos^{-1}(1/x)\), etc.
The Law of Sines
Solving Oblique Triangles
An oblique triangle has no right angle. A triangle is determined by three of its six parts, as long as at least one is a side. The four cases: ASA/SAA (one side, two angles), SSA (two sides and a non-included angle), SAS (two sides and included angle), SSS (three sides). The Law of Sines handles Cases 1 and 2; the Law of Cosines handles Cases 3 and 4.
In any triangle the sides are proportional to the sines of their opposite angles (angles \(A,B,C\) opposite sides \(a,b,c\)).
The Ambiguous Case (SSA)
In the SSA case there may be two triangles, one triangle, or no triangle. After solving for \(\sin B\): if \(\sin B>1\), no triangle exists. If \(\sin B\le 1\), check both the angle and its supplement \(180^\circ-B\) — keep each only if the resulting angle sum stays under \(180^\circ\). When both work, two distinct triangles solve the problem.
Section 6.5 note
The ambiguous case is where careless students lose marks. My rule: only use Law of Sines to find an angle you already suspect is acute (the one opposite the shorter known side). If you must find a potentially obtuse angle, switch to Law of Cosines (6.6) — its result is unique. In surveying and structural layout, missing the second valid triangle can put a member in the wrong place entirely.
The Law of Cosines
When to Use It
The Law of Sines cannot directly solve a triangle when we know two sides and the included angle (SAS) or all three sides (SSS). The Law of Cosines covers exactly these cases. If one angle is \(90^\circ\), \(\cos C=0\) and the law reduces to the Pythagorean Theorem — so the Pythagorean Theorem is a special case.
The square of any side equals the sum of the squares of the other two minus twice their product times the cosine of the included angle.
Navigation: Heading and Bearing
A bearing is an acute angle measured from due north or due south toward east or west. For example, N 30° E points \(30^\circ\) east of due north. Bearings convert a navigation problem into a triangle that the Laws of Sines and Cosines can solve.
Here \(s\) is the semiperimeter (half the perimeter).
Section 6.6 note
Prefer the Law of Cosines whenever you can — every angle from \(0^\circ\) to \(180^\circ\) has a unique cosine, so there's no ambiguous case to trip over. In a SSS or SAS solve, find the largest angle first (opposite the longest side); if its cosine comes out negative, you've correctly caught an obtuse triangle. Heron's formula is your friend for irregular plates and gussets where you only have edge lengths.
Surveying
Measuring Distances from Angles
Distances are often hard to measure directly, but angles to distant objects are easy. Surveying exploits this: in triangulation, a single baseline distance is measured, then a chain of interlocking triangles is built up — every other distance is computed from measured angles using the Law of Sines. This is the same principle by which the Great Trigonometric Survey of India fixed the height of Everest.
Focus on Modeling note
Triangulation is the conceptual ancestor of how GPS and total stations work today. The takeaway for an engineer: a network is only as accurate as its single measured baseline plus the angular precision — errors propagate through every dependent triangle. When you trust a CAD survey import, remember a human (or instrument) once measured exactly one length and a pile of angles.