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Trig Identities, Equations, Laws & Vectors — Visual Lab
Module 4. Trigonometry leaves the unit circle and goes to work: it solves any triangle (not just right ones) and it gives direction to quantities that have size and heading. Drag the figure and watch the Law of Sines, the Law of Cosines, and head-to-tail vector addition come alive on the coordinate plane.
First we verify and simplify with identities and solve trig equations — remembering the whole \(+2\pi n\) family of answers. Then we point trig at real triangles with the Law of Sines and Law of Cosines (and stare down the ambiguous SSA case), and finally model magnitude-and-direction with vectors. Drag the lab below and keep the figure, the algebra, and the plain-English readout in perfect agreement.
The big ideas, stated precisely
Four moves define this module. Read them once before you play, then watch the lab make them concrete.
Rewrite, then solve
The Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) and its cousins let you swap one expression for an equal one until a trig equation simplifies. When you solve, the periodicity of sine and cosine means every solution repeats: write the whole family
\[ \theta = \theta_0 + 2\pi n,\quad n \in \mathbb{Z}. \]
And never divide both sides by a trig factor — factor instead, or you'll erase solutions where that factor is zero.
Any triangle; any direction
For a triangle with sides \(a,b,c\) opposite angles \(A,B,C\):
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \qquad c^2 = a^2 + b^2 - 2ab\cos C. \]
A vector \(\langle x, y\rangle\) carries magnitude \(\sqrt{x^2+y^2}\) and a direction angle. You add vectors head-to-tail — by adding components, never by adding magnitudes.
Triangle & Vector Lab
Pick a mode, then drag the labelled dots (or use the keyboard pad). In Triangle, move A, B, C and read the live Law-of-Sines / Law-of-Cosines values, plus run an SSA solve with an ambiguous-case warning. In Vector Addition, drag the arrowheads of \(\mathbf{u}\) and \(\mathbf{v}\), scale \(\mathbf{v}\), and watch them add head-to-tail into a resultant.
What you're seeing
- Each side is named for the angle across from it. Side \(a\) faces \(A\), \(b\) faces \(B\), \(c\) faces \(C\). The lab measures all three sides and all three angles as you drag.
- The Law of Sines is a single shared ratio. The readout shows \(a/\sin A = b/\sin B = c/\sin C\) collapsing to one number — the diameter of the triangle's circumscribed circle.
- The Law of Cosines links a side to its opposite angle. Watch \(c^2 = a^2 + b^2 - 2ab\cos C\) stay true for every shape, even obtuse ones.
- The SSA panel warns about the ambiguous case. Given two sides and a non-included angle, there can be zero, one, or two triangles. The box tells you which, and why.
- Vectors add head-to-tail. The dashed copy of \(\mathbf{v}\) is slid onto the tip of \(\mathbf{u}\); its arrowhead lands exactly on the dark resultant \(\mathbf{u}+\mathbf{v}\). The readout proves the magnitudes don't simply add.
Try this
- Drag C until the angle sum reads exactly 180°. (It always does — the readout proves the angle-sum theorem no matter how you distort the triangle.)
- Make one angle obtuse. Notice that \(\cos C\) goes negative in the Law of Cosines, so \(c^2\) grows larger than \(a^2+b^2\). The longest side always faces the largest angle.
- In the SSA panel, set \(A = 35^\circ\), \(a = 7\), \(b = 9\). The warning flips to two triangles — find the value of \(a\) (just below \(b\)) where it switches to one, then to none.
- Switch to Vector mode and point \(\mathbf{u}\) and \(\mathbf{v}\) the same way. Only now does the resultant magnitude equal the sum of the magnitudes. Tilt one away and watch the gap open.
- Drag the scalar to \(-1\). The vector \(\mathbf{v}\) reverses to \(-\mathbf{v}\): same length, opposite direction — that's scalar multiplication, geometrically.
Worked examples
Two complete solutions — one solving the ambiguous triangle, one adding vectors — modeled the way you should write them.
In \(\triangle ABC\), \(A = 35^\circ\), \(a = 7\), \(b = 9\). Solve the triangle.
- Check the height first. The altitude from \(C\) to side \(c\) is \(h = b\sin A = 9\sin 35^\circ \approx 5.16\). Since \(h < a < b\) ( \(5.16 < 7 < 9\) ), two triangles are possible — this is the ambiguous case.
- Law of Sines for \(\sin B\). \(\dfrac{\sin B}{b} = \dfrac{\sin A}{a}\), so \(\sin B = \dfrac{9\sin 35^\circ}{7} \approx 0.7373\).
- Take both angles with that sine. \(B_1 = \sin^{-1}(0.7373) \approx 47.5^\circ\) and \(B_2 = 180^\circ - 47.5^\circ \approx 132.5^\circ\). Both keep \(A + B < 180^\circ\), so both survive.
- Find each third angle, then side \(c\). Triangle 1: \(C_1 = 180^\circ - 35^\circ - 47.5^\circ = 97.5^\circ\); Triangle 2: \(C_2 = 180^\circ - 35^\circ - 132.5^\circ = 12.5^\circ\). Then \(c = \dfrac{a\sin C}{\sin A}\).
Let \(\mathbf{u} = \langle 5, 1\rangle\) and \(\mathbf{v} = \langle 2, 4\rangle\). Find \(\mathbf{u}+\mathbf{v}\), its magnitude, and its direction angle.
- Add componentwise (not the lengths). \(\mathbf{u}+\mathbf{v} = \langle 5+2,\ 1+4\rangle = \langle 7, 5\rangle\).
- Magnitude with the distance formula. \(\lVert \mathbf{u}+\mathbf{v}\rVert = \sqrt{7^2 + 5^2} = \sqrt{74} \approx 8.60\).
- Direction from the components. \(\theta = \tan^{-1}\!\left(\dfrac{5}{7}\right) \approx 35.5^\circ\) (first quadrant, so no adjustment needed).
- Sanity check the "don't add lengths" trap. \(\lVert\mathbf{u}\rVert + \lVert\mathbf{v}\rVert = \sqrt{26} + \sqrt{20} \approx 9.57\), which is not \(8.60\). Components win.
From the unit circle to the real world
Right-triangle trig only handles right triangles — but surveyors, navigators, and engineers face triangles of every shape. The Law of Sines and Law of Cosines are how you find an unreachable distance from two angles and a baseline, or a bearing from three known points. Vectors are the language of force, velocity, and displacement: a plane's true course is its airspeed vector plus the wind vector, added head-to-tail. Master both here and you have the toolkit physics and calculus assume you already own — this is the edge of trigonometry where it stops being about circles and starts being about the world.
Key Vocabulary
The precise words a mathematician uses to describe what the lab is doing.
An equation true for every value of the variable, e.g. \(\sin^2\theta + \cos^2\theta = 1\). Used to rewrite and simplify expressions.
All solutions of a trig equation, written with the period: \(\theta = \theta_0 + 2\pi n\) for integer \(n\) (or \(+\,360^\circ n\) in degrees).
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\). Best for AAS, ASA, and SSA setups (a side / opposite-angle pair).
\(c^2 = a^2 + b^2 - 2ab\cos C\). Best for SAS and SSS — when no side / opposite-angle pair is known.
Two sides and a non-included angle may determine zero, one, or two triangles. Compare side \(a\) to the altitude \(h = b\sin A\) to decide.
A quantity \(\langle x,y\rangle\) with magnitude \(\sqrt{x^2+y^2}\) and direction. The resultant of a sum is found head-to-tail, by adding components.
TEKS §111.42 (Precalculus)
This lab targets the trigonometric laws and vector strands of Module 4, where scholars solve any triangle and model magnitude-and-direction quantities, alongside the identity and equation work of P.5.
P.4(G) apply the Law of Sines · P.4(H) apply the Law of Cosines · P.4(I) represent the magnitude and direction of a vector · P.4(J) represent vector addition and scalar multiplication geometrically and symbolically · P.4(K) apply vector addition and multiplication by a scalar in mathematical and real-world problems · P.5(M) reciprocal, quotient, Pythagorean, cofunction, even-odd, and sum-difference identities · P.5(N) solve trigonometric equations.
Shaky on the basics? Build them first on Module 4 Foundations. Ready for the full course map? Head back to Pre-Calculus, read the Student Support, or check the Pacing Guide to see where Module 4 sits in the year.
Module and topic structure follow the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Classroom use is non-commercial.