Trigonometry

Connecting an angle to the sides of a right-angled triangle — which makes heights and distances you cannot reach into things you can calculate.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Geometry and Measure)ตรงกับ ม.4 อัตราส่วนตรีโกณมิติ — หลักสูตรไทยสอนช้ากว่าหนึ่งปี
Common Core (US): G-SRT
Cambridge: Lower Secondary Stage 9 — Geometry and Measure: Pythagoras and trigonometry

9.1 Sine, cosine and tangent

The three ratios

sin θ = \frac{opposite}{hypotenuse} cos θ = \frac{adjacent}{hypotenuse} tan θ = \frac{opposite}{adjacent}

Opposite and adjacent are named relative to the angle you have chosen — they are not fixed properties of the triangle.

ข้อควรระวัง

A side that is adjacent to one acute angle is opposite the other. Mark the angle you are working with FIRST, then label the sides. Labelling first is how $sin$ and $cos$ get swapped.

Choosing the right ratio

1Mark the angle you are using
2Label the three sides relative to it: opposite, adjacent, hypotenuse
3Note which two sides the question involves — the one you know and the one you want
4Pick the ratio containing exactly those two

ตัวอย่าง

A ladder leans against a wall at $6 5^{\circ}$ to the ground, with its foot $2.4$ m from the wall. How long is the ladder?

วิธีทำ

1.The $2.4$ m is adjacent to the $6 5^{\circ}$ angle; the ladder is the hypotenuse
2.Adjacent and hypotenuse means cosine
3. $cos 6 5^{\circ} = \frac{2.4}{L}$
4. $L = \frac{2.4}{cos 6 5 ^{\circ}} \approx \frac{2.4}{0.4226}$

ตอบ About $5.7$ m

ข้อควรระวัง

Check the calculator is in DEGREES, not radians. A radian-mode answer looks like a perfectly ordinary number and is simply wrong, with nothing on the page to suggest it.

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