7.4 Right Triangle Trigonometry & Radian Measure

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.

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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:

1. 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: $$BC = 2 \times 5 = 10$$
2. Evaluate $\cos(A)$: Side adjacent to angle $A$ is $AC = 24$; hypotenuse is $AB = 26$: $$\cos(A) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{24}{26} = \frac{12}{13}$$
3. Evaluate $\tan(B)$: Side opposite to angle $B$ is $AC = 24$; side adjacent to angle $B$ is $BC = 10$: $$\tan(B) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{24}{10} = \frac{12}{5}$$

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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:

1. Apply the co-function rule: Because $\sin(A) = \cos(B)$, the angle expressions must sum to $90^\circ$: $$(4x - 10) + (x + 25) = 90$$
2. Combine like terms and solve for $x$: $$5x + 15 = 90 \implies 5x = 75 \implies x = 15$$

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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

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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:

1. Use the fundamental Pythagorean Identity: $$\sin^2\theta + \cos^2\theta = 1 \implies \sin^2\theta + \left(\frac{7}{25}\right)^2 = 1$$
2. Solve for $\sin\theta$: $$\sin^2\theta + \frac{49}{625} = 1 \implies \sin^2\theta = 1 - \frac{49}{625} = \frac{576}{625}$$ $$\sin\theta = \sqrt{\frac{576}{625}} = \frac{24}{25}\text{ (positive in Quadrant I)}$$
3. Compute $\tan\theta$: $$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{24/25}{7/25} = \frac{24}{7}$$