Right triangles and trigonometry

Right Triangles and Trigonometry

Right triangles ($90^\circ$ interior angle) form a cornerstone of SAT geometry. Their side lengths and acute angle relationships can be solved using the Pythagorean theorem, special right triangle shortcuts, or basic trigonometric ratios.

• Pythagorean Theorem: For any right triangle with legs $a$ and $b$ and hypotenuse $c$:

  $$a^2 + b^2 = c^2$$

• Common Pythagorean Triples: Memorizing these side length ratios saves critical time:

  • $3 : 4 : 5$ (and multiples like $6 : 8 : 10$ or $9 : 12 : 15$)

  • $5 : 12 : 13$

  • $8 : 15 : 17$

  • $7 : 24 : 25$

Special Right Triangles

When specific acute angles are present, you can determine all side lengths using exact proportional ratios without running the Pythagorean formula.

• $45^\circ-45^\circ-90^\circ$ Triangle (Isosceles Right):

  • Angle Ratio: $45^\circ : 45^\circ : 90^\circ$

  • Side Ratio: $x : x : x\sqrt{2}$

  • Both legs are equal ($x$), and the hypotenuse is the leg length multiplied by $\sqrt{2}$.

• $30^\circ-60^\circ-90^\circ$ Triangle:

  • Angle Ratio: $30^\circ : 60^\circ : 90^\circ$

  • Side Ratio: $x : x\sqrt{3} : 2x$

  • The leg opposite $30^\circ$ is the shortest side ($x$), the leg opposite $60^\circ$ is $x\sqrt{3}$, and the hypotenuse opposite $90^\circ$ is $2x$.

Right Triangle Trigonometry (SOH CAH TOA)

For an acute angle $\theta$ in a right triangle:

• Sine: $\sin(\theta) = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$

• Cosine: $\cos(\theta) = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$

• Tangent: $\tan(\theta) = \dfrac{\text{Opposite}}{\text{Adjacent}} = \dfrac{\sin(\theta)}{\cos(\theta)}$

Sine and Cosine Complementary Angle Identity

In any right triangle, the two acute angles sum to $90^\circ$ (they are complementary). If $\angle A + \angle B = 90^\circ$:

1. Use Special Right Ratios to Avoid Calculation. If you identify a $30^\circ-60^\circ-90^\circ$ or $45^\circ-45^\circ-90^\circ$ angle, set up side ratios immediately. 

   Example: A $30^\circ-60^\circ-90^\circ$ triangle has a short leg ($x$) of $5$. Find the remaining sides.

   • Short Leg ($30^\circ$): $x = 5$

   • Long Leg ($60^\circ$): $x\sqrt{3} = 5\sqrt{3}$

   • Hypotenuse ($90^\circ$): $2x = 2(5) = 10$

2. Use Complementary Angle Identities for Trigonometric Equations. When the SAT sets $\sin(x) = \cos(y)$, the angles inside must add up to $90^\circ$. 

   Example: If $\sin(2x + 10^\circ) = \cos(3x - 5^\circ)$, find the value of $x$.

   • Set the angle expressions' sum equal to $90^\circ$:

     $$(2x + 10) + (3x - 5) = 90$$

   • Simplify and solve:

     $$5x + 5 = 90 \implies 5x = 85 \implies x = 17$$

3. Apply SOH CAH TOA to Find Missing Side Lengths. Set up a simple trigonometric equation when given one angle and one side length. 

   Example: In right triangle $\triangle ABC$ with right angle at $C$, angle $A = 30^\circ$ and hypotenuse $AB = 12$. Find length $BC$ (the side opposite $\angle A$).

   • Set up the sine definition:

     $$\sin(30^\circ) = \dfrac{\text{Opposite}}{\text{Hypotenuse}} = \dfrac{BC}{12}$$

   • Substitute $\sin(30^\circ) = \dfrac{1}{2}$:

     $$\dfrac{1}{2} = \dfrac{BC}{12} \implies BC = 6$$