SAT-75: Right Triangle Trigonometry — Sine, Cosine, Tangent

SAT-75: Right Triangle Trigonometry — Sine, Cosine, Tangent

Description: Trigonometry connects a right triangle's angles to its side ratios. The three core functions — sine, cosine, tangent — are captured by one memory phrase, SOH-CAH-TOA.

Naming the sides (from a chosen angle)

Pick the angle you care about (not the right angle). Then:

• Opposite = the side across from that angle.
• Adjacent = the side next to that angle (not the hypotenuse).
• Hypotenuse = the longest side, opposite the right angle.

(Oʻzbekcha: tanlangan burchakka qarab tomonlar nomlanadi: qarshi, yondosh va gipotenuza.)

SOH-CAH-TOA

sin = Opposite / Hypotenuse
cos = Adjacent / Hypotenuse
tan = Opposite / Adjacent

(Oʻzbekcha: sinus = qarshi/gipotenuza, kosinus = yondosh/gipotenuza, tangens = qarshi/yondosh.)

Worked Example 1 — find a ratio

A right triangle has opposite = 3, adjacent = 4, hypotenuse = 5 (relative to angle θ). Find sin θ, cos θ, tan θ.

• sin θ = 3/5, cos θ = 4/5, tan θ = 3/4.

Worked Example 2 — find a missing side

In a right triangle, the angle is 30° and the hypotenuse is 12. Find the side opposite the 30°.

• Opposite uses sine: sin 30° = opposite / 12. Since sin 30° = 0.5, opposite = 0.5 × 12 = 6.

(Oʻzbekcha: qarshi tomonni topish uchun sinusdan foydalanamiz.)

Worked Example 3 — choose the right function

You know the angle 40° and the adjacent side 10, and want the opposite side. Which function?

• Opposite and adjacent together → tangent: tan 40° = opposite / 10, so opposite = 10·tan 40°.

How to choose: see which two sides are involved (the one you have and the one you want), then pick the function that uses exactly those two. (Oʻzbekcha: qaysi ikki tomon ishtirok etsa, oʻsha funksiyani tanlang.)

Worked Example 4 — find the angle (inverse trig)

A right triangle has opposite = 6 and adjacent = 6 for angle θ. What is θ?

• tan θ = opposite/adjacent = 6/6 = 1.
• The angle whose tangent is 1 is 45° (use the inverse: θ = tan⁻¹(1)).

When you know a ratio and want the angle, you use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹. (Oʻzbekcha: nisbatdan burchakni topish uchun teskari funksiyalardan (tan⁻¹) foydalanamiz.)

The famous angle values to know

A few values come up so often they are worth memorizing: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = √2/2; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. These match the special triangles from SAT-71 and SAT-72, so you are really just reading side ratios off those triangles. (Oʻzbekcha: 30°, 45°, 60° qiymatlari maxsus uchburchaklardan kelib chiqadi — ularni yodlang.)

Common mistake — wrong side as hypotenuse

Sine and cosine always divide by the hypotenuse, never by a leg. If you accidentally use a leg as the denominator, every ratio comes out wrong. Identify the hypotenuse (opposite the right angle, longest side) before writing any ratio. (Oʻzbekcha: sinus va kosinus doim gipotenuzaga boʻlinadi — avval gipotenuzani aniqlang.)

Practice 1

A right triangle has opposite = 5 and hypotenuse = 13 for angle θ. Find sin θ and cos θ.

Practice 2

An angle is 60° and its adjacent side is 8. Find the opposite side using tangent (tan 60° = √3).

Key words — Kalit soʻzlar

• Trigonometry — trigonometriya
• Sine — sinus
• Cosine — kosinus
• Tangent — tangens
• Opposite side — qarshi tomon
• Adjacent side — yondosh tomon
• Hypotenuse — gipotenuza
• Ratio — nisbat
• Angle — burchak

Summary

• From a chosen angle, label sides opposite / adjacent / hypotenuse.
• SOH-CAH-TOA: sin = O/H, cos = A/H, tan = O/A.
• Pick the function that uses the side you have and the side you want.