SAT-76: Trigonometric Identities — sin(x) = cos(90° − x)

SAT-76: Trigonometric Identities — sin(x) = cos(90° − x)

Description: The SAT often tests one elegant fact: the sine of an angle equals the cosine of its complement. This is the cofunction identity, and it follows directly from SOH-CAH-TOA.

The identity

sin(x) = cos(90° − x) and cos(x) = sin(90° − x)

Because the two non-right angles of a right triangle add to 90°, they are complementary — so one angle's sine is the other angle's cosine. (Oʻzbekcha: toʻgʻri burchakli uchburchakda ikki oʻtkir burchak 90° ga toʻldiradi, shuning uchun biri sinusi ikkinchisining kosinusiga teng.)

Why it is true (from the triangle)

For angle x, the opposite side is the adjacent side for angle (90° − x), and vice versa. So sin x = opposite/hyp for x is exactly cos(90° − x) = adjacent/hyp for the other angle. Same ratio, two names. (Oʻzbekcha: bir burchakning qarshi tomoni — ikkinchi burchakning yondosh tomoni, shuning uchun nisbatlar mos keladi.)

Worked Example 1 — direct substitution

If sin(20°) = 0.34, what is cos(70°)?

• 70° = 90° − 20°, so cos(70°) = sin(20°) = 0.34.

Worked Example 2 — match the angles

cos(35°) equals the sine of what angle?

• cos(35°) = sin(90° − 35°) = sin(55°).

(Oʻzbekcha: kosinusni sinusga aylantirish uchun burchakni 90° dan ayiramiz.)

Worked Example 3 — solve for an unknown angle

If sin(2x) = cos(3x), and the angles are acute, find x.

• sin(2x) = cos(3x) means the angles are complementary: 2x + 3x = 90.
• 5x = 90 → x = 18°.

Key move: whenever you see "sin(A) = cos(B)" with acute angles, set A + B = 90°. (Oʻzbekcha: sin(A) = cos(B) boʻlsa, A + B = 90° deb yozing.)

Worked Example 4 — same value, find the partner angle

For what acute angle is the sine equal to cos(28°)?

• cos(28°) = sin(90° − 28°) = sin(62°), so the angle is 62°.
• Notice 28° and 62° add to 90° — they are the two acute angles of the same right triangle.

(Oʻzbekcha: 28° va 62° — bir uchburchakning ikki oʻtkir burchagi.)

Why this saves time on the SAT

Many questions look harder than they are because they mix sine and cosine. The cofunction identity lets you rewrite one in terms of the other so both sides speak the same "language." Once both are sines (or both cosines), you can match angles directly. Train yourself to spot the pattern: a sine and a cosine set equal, with angles that could add to 90°, is almost always a cofunction question. (Oʻzbekcha: sinus va kosinus aralashgan masalalarni kofunksiya yordamida bitta turga keltiring.)

A note on equal sines

Be careful: sin(A) = sin(B) does not force A = B in general, but for the acute angles the SAT uses, equal sines do mean equal angles, and equal cosines mean equal angles. The cofunction twist only appears when one is sine and the other is cosine. (Oʻzbekcha: oʻtkir burchaklar uchun teng sinuslar teng burchaklarni bildiradi; kofunksiya esa sinus-kosinus aralashganda kerak.)

Practice 1

If cos(25°) = 0.91, what is sin(65°)?

Practice 2

If sin(x + 10°) = cos(x + 30°) with acute angles, find x.

Key words — Kalit soʻzlar

• Identity — ayniyat
• Cofunction — kofunksiya
• Complementary angles — toʻldiruvchi burchaklar (90°)
• Sine / Cosine — sinus / kosinus
• Acute angle — oʻtkir burchak
• Substitute — oʻrniga qoʻyish
• Equal ratios — teng nisbatlar
• Right triangle — toʻgʻri burchakli uchburchak
• Sum to 90° — 90° ga toʻldirish

Summary

• Cofunction identity: sin(x) = cos(90° − x) (and cos(x) = sin(90° − x)).
• It works because the two acute angles of a right triangle are complementary.
• For "sin(A) = cos(B)" with acute angles, set A + B = 90° and solve.