7.4 Right Triangle Trigonometry & Radian Measure
Key Takeaways
- Trigonometric ratios in right triangles are defined by SOH-CAH-TOA: $\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$, $\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$, and $\tan\theta = \frac{\text{Opp}}{\text{Adj}} = \frac{\sin\theta}{\cos\theta}$.
- The Complementary Angle Identity states that the sine of an acute angle equals the cosine of its complement: $\sin(x) = \cos(90^\circ - x)$ or in radians $\sin(x) = \cos\left(\frac{\pi}{2} - x\right)$.
- Radian measure defines angle size by arc length on a unit circle ($2\pi\text{ rad} = 360^\circ$); convert via $\text{rad} = \text{deg} \times \frac{\pi}{180^\circ}$ and $\text{deg} = \text{rad} \times \frac{180^\circ}{\pi}$.
- On the unit circle, any terminal point has coordinates $(x, y) = (\cos\theta, \sin\theta)$, yielding the fundamental Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$.
- On the Digital SAT Desmos calculator, verify whether the wrench settings are set to DEG (Degrees) or RAD (Radians) before evaluating trigonometric expressions.
7.4 Right Triangle Trigonometry & Radian Measure
Trigonometry on the Digital SAT Math section is focused entirely on right triangle definitions, complementary angle relationships, radian conversions, and foundational unit circle coordinates. The exam does not test advanced precalculus identities (such as sum/difference formulas or double-angle formulas). Mastering the core definitions and co-function relationships ensures rapid, high-accuracy problem solving.
1. SOH-CAH-TOA Definitions
In any right triangle with acute reference angle $\theta$:
+-----------------------------------------------------------------------------+
| SOH - CAH - TOA DEFINITIONS |
| |
| /| |
| / | |
| Hypotenuse / | Opposite Side |
| (H) / | (O) |
| / θ | |
| +-----+ |
| Adjacent Side (A) |
| |
| [ SOH ] sin(θ) = Opposite / Hypotenuse = O / H |
| [ CAH ] cos(θ) = Adjacent / Hypotenuse = A / H |
| [ TOA ] tan(θ) = Opposite / Adjacent = O / A = sin(θ) / cos(θ) |
+-----------------------------------------------------------------------------+
Key Invariance Principle
Trigonometric ratios depend only on the angle measure $\theta$, NOT on the physical size of the triangle. Because all right triangles containing acute angle $\theta$ are similar by AA similarity, their side ratios are strictly constant.
Worked Example 1: Finding Trig Ratios from Side Lengths
Problem: In right triangle $\Delta ABC$ with right angle at $C$, the length of leg $AC = 24$ and the length of hypotenuse $AB = 26$. What is the value of $\cos(A)$ and $\tan(B)$?
Step-by-Step Solution:
- Identify the missing leg $BC$ using Pythagorean triples: Notice $24 = 2 \times 12$ and $26 = 2 \times 13$. This is the $(5, 12, 13)$ triple scaled by 2:
- Evaluate $\cos(A)$: Side adjacent to angle $A$ is $AC = 24$; hypotenuse is $AB = 26$:
- Evaluate $\tan(B)$: Side opposite to angle $B$ is $AC = 24$; side adjacent to angle $B$ is $BC = 10$:
2. The Complementary Angle (Co-Function) Identity
In any right triangle, the two acute angles $A$ and $B$ are complementary ($A + B = 90^\circ$ or $A + B = \frac{\pi}{2}\text{ rad}$).
+-----------------------------------------------------------------------------+
| THE CO-FUNCTION COMPLEMENTARY IDENTITY |
| |
| sin(x) = cos(90° - x) |
| cos(x) = sin(90° - x) |
| |
| In Radians: sin(x) = cos(π/2 - x) |
| cos(x) = sin(π/2 - x) |
| |
| CORE RULE: If sin(A) = cos(B), then: |
| A + B = 90° (in degrees) OR A + B = π/2 (in radians) |
+-----------------------------------------------------------------------------+
Why This Works Geometrically
Looking at $\Delta ABC$ with right angle $C$:
- $\sin(A) = \frac{\text{side opposite } A}{\text{hypotenuse}} = \frac{a}{c}$
- $\cos(B) = \frac{\text{side adjacent } B}{\text{hypotenuse}} = \frac{a}{c}$
Because the side opposite angle $A$ is the exact same side adjacent to angle $B$, $\sin(A)$ and $\cos(B)$ are identical.
Worked Example 2: Algebraic Co-Function Equation
Problem: If $\sin(4x - 10^\circ) = \cos(x + 25^\circ)$ where both angles are acute, what is the value of $x$?
Step-by-Step Solution:
- Apply the co-function rule: Because $\sin(A) = \cos(B)$, the angle expressions must sum to $90^\circ$:
- Combine like terms and solve for $x$:
[!IMPORTANT] The Most Tested Trig Question on the SAT: Whenever the College Board sets a sine expression equal to a cosine expression ($\sin(\text{Expr}_1) = \cos(\text{Expr}_2)$), immediately set $\text{Expr}_1 + \text{Expr}_2 = 90^\circ$ (or $\frac{\pi}{2}$). You do not need to evaluate the sine or cosine values.
3. Radian Measure and Degree Conversions
A radian is the standard unit of angular measure in calculus and higher mathematics. One radian is the measure of a central angle that subtends an arc equal in length to the circle's radius.
+-----------------------------------------------------------------------------+
| DEGREE <---> RADIAN CONVERSION |
| |
| Equivalence: 360° = 2π radians <===> 180° = π radians |
| |
| [ Degrees to Radians ] [ Radians to Degrees ] |
| Multiply by: π / 180° Multiply by: 180° / π |
| |
| Example: 60° * (π/180°) = π/3 rad Example: (3π/4) * (180°/π) = 135° |
+-----------------------------------------------------------------------------+
Benchmark Angle Conversions Table
| Degrees ($^\circ$) | Radians (rad) | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ |
|---|---|---|---|---|
| $0^\circ$ | $0$ | $0$ | $1$ | $0$ |
| $30^\circ$ | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ |
| $45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
| $60^\circ$ | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| $90^\circ$ | $\frac{\pi}{2}$ | $1$ | $0$ | Undefined |
| $180^\circ$ | $\pi$ | $0$ | $-1$ | $0$ |
| $270^\circ$ | $\frac{3\pi}{2}$ | $-1$ | $0$ | Undefined |
| $360^\circ$ | $2\pi$ | $0$ | $1$ | $0$ |
4. Unit Circle Coordinates & Pythagorean Identity
The unit circle is a circle of radius $r = 1$ centered at the origin $(0, 0)$ in the $xy$-plane.
+-----------------------------------------------------------------------------+
| THE UNIT CIRCLE & QUADRANTS |
| |
| y (0, 1) |
| | |
| Quadrant II | Quadrant I |
| (-x, +y) | (+x, +y) |
| SIN is (+) | ALL are (+) |
| | |
| (-1, 0) -----------------+----------------- (1, 0) x |
| | |
| Quadrant III | Quadrant IV |
| (-x, -y) | (+x, -y) |
| TAN is (+) | COS is (+) |
| | |
| (0, -1) |
| |
| Point on Unit Circle: (x, y) = (cos(θ), sin(θ)) |
| Pythagorean Identity: cos²(θ) + sin²(θ) = 1 |
| Mnemonic for Signs: "All Students Take Calculus" (ASTC: Q1-Q2-Q3-Q4) |
+-----------------------------------------------------------------------------+
Worked Example 3: Applying the Pythagorean Identity
Problem: If angle $\theta$ is in Quadrant I and $\cos\theta = \frac{7}{25}$, what is the value of $\sin\theta$ and $\tan\theta$?
Step-by-Step Solution:
- Use the fundamental Pythagorean Identity:
- Solve for $\sin\theta$:
- Compute $\tan\theta$:
[!TIP] Desmos Calculator Mode Check: On the Digital SAT, open the Wrench icon in the top-right corner of Desmos. Ensure the angle toggle is highlighted on Degrees or Radians matching the units specified in the question.
For acute angle measures, if sin(5x - 12°) = cos(3x + 6°), what is the value of x?
In right triangle ΔDEF, the right angle is at vertex F, and sin(D) = 7/25. What is the value of tan(E)?
An angle measures 7π/4 radians. What is the equivalent measure of this angle in degrees, and what is the value of cos(7π/4)?