6. Trigonometric Ratios

Trigonometry extends right-triangle relationships to find missing sides and angles using three primary ratios. For an acute angle theta in a right triangle:

SOH-CAH-TOA:

• sin(theta) = Opposite / Hypotenuse
• cos(theta) = Adjacent / Hypotenuse
• tan(theta) = Opposite / Adjacent

Finding a missing side: Identify which sides are involved and which trig ratio connects them, then solve.
Example: A ladder leans against a wall at a 60-degree angle. The ladder is 12 feet long. How high on the wall does it reach?

• The height is opposite to 60 degrees; the ladder (12) is the hypotenuse.
• sin(60) = height / 12, so height = 12 sin(60) = 12 (sqrt(3)/2) = 6*sqrt(3).

Key angle values to memorize:

• sin(30) = 1/2, cos(30) = sqrt(3)/2, tan(30) = 1/sqrt(3) = sqrt(3)/3
• sin(45) = cos(45) = sqrt(2)/2, tan(45) = 1
• sin(60) = sqrt(3)/2, cos(60) = 1/2, tan(60) = sqrt(3)

Complementary angle identity: sin(theta) = cos(90 - theta). On the SAT®, an equation like sin(x) = cos(32) means x + 32 = 90, so x = 58.

Radians vs. degrees: A full circle is 360 degrees or 2*pi radians.

• Conversion: degrees pi / 180 = radians; radians 180 / pi = degrees.
• Common conversions: 30 = pi/6, 45 = pi/4, 60 = pi/3, 90 = pi/2, 180 = pi.

Worked example: In right triangle ABC, angle C = 90 degrees, AB = 13, BC = 5. Find sin(A) and cos(A).

• BC is opposite angle A; AB is the hypotenuse.
• sin(A) = opposite/hypotenuse = 5/13.
• The third side AC = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12.
• cos(A) = adjacent/hypotenuse = 12/13.

• Pythagorean identity: sin^2(theta) + cos^2(theta) = 1 -- useful for substitution problems.
• Law of Sines / Cosines: Not typically required on the SAT®, but complementary identity questions appear frequently.

⚠️ Common mistake: Mixing up opposite and adjacent. Always label the triangle relative to the angle in question -- the opposite side never touches the angle; the adjacent side does (and is not the hypotenuse).

💡 Tip: If you forget a key trig value during the test, quickly sketch a 30-60-90 or 45-45-90 triangle using the special-right-triangle ratios from the formula sheet and read off the ratio directly.