Sine, cosine, and tangent in a right triangle

Key idea

In a right triangle, pick one acute angle \(\theta\). Relative to \(\theta\): opposite is across from \(\theta\), adjacent touches \(\theta\) (but is not the hypotenuse), and the hypotenuse is across from the \(90^\circ\) angle. Then: \[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}. \] This is the classic mnemonic SOHCAHTOA.

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Summary

• SOHCAHTOA: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\).
• Opposite/adjacent depend on which acute angle you call \(\theta\).

Secant, cosecant, and cotangent

Key idea

The reciprocal trig ratios are: \[ \csc\theta=\frac{1}{\sin\theta}=\frac{\text{hyp}}{\text{opp}},\quad \sec\theta=\frac{1}{\cos\theta}=\frac{\text{hyp}}{\text{adj}},\quad \cot\theta=\frac{1}{\tan\theta}=\frac{\text{adj}}{\text{opp}}. \] If you can compute \(\sin\), \(\cos\), or \(\tan\), you can get \(\csc\), \(\sec\), or \(\cot\) immediately by flipping the fraction.

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Summary

• \(\sec\theta=\dfrac{\text{hyp}}{\text{adj}}\), \(\csc\theta=\dfrac{\text{hyp}}{\text{opp}}\), \(\cot\theta=\dfrac{\text{adj}}{\text{opp}}\).
• Reciprocal rule: \(\sec=\dfrac{1}{\cos}\), \(\csc=\dfrac{1}{\sin}\), \(\cot=\dfrac{1}{\tan}\).

Find missing sides, then compute ratios

Key idea

Many right-triangle ratio problems are a two-step process: (1) find a missing side using the Pythagorean theorem \(a^2+b^2=c^2\), then (2) compute \(\sin\theta\), \(\cos\theta\), \(\tan\theta\) or their reciprocals. If you’re given a ratio like \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\), you can often reconstruct the triangle.

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Summary

• Use \(a^2+b^2=c^2\) to find missing sides when needed.
• Then compute the desired trig ratio using SOHCAHTOA or reciprocals.

Special triangles and exact trig values

Key idea

Two special right triangles give exact trig values: \(45\text{-}45\text{-}90\) has side ratio \(1:1:\sqrt{2}\), and \(30\text{-}60\text{-}90\) has side ratio \(1:\sqrt{3}:2\). From these, you can derive: \[ \sin 30^\circ=\frac12,\ \cos 30^\circ=\frac{\sqrt3}{2},\ \tan 30^\circ=\frac{\sqrt3}{3} \] \[ \sin 45^\circ=\frac{\sqrt2}{2},\ \cos 45^\circ=\frac{\sqrt2}{2},\ \tan 45^\circ=1 \] \[ \sin 60^\circ=\frac{\sqrt3}{2},\ \cos 60^\circ=\frac12,\ \tan 60^\circ=\sqrt3 \]

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Summary

• Special triangles give exact values at \(30^\circ\), \(45^\circ\), \(60^\circ\).
• Use reciprocals to get \(\sec\), \(\csc\), \(\cot\) from \(\cos\), \(\sin\), \(\tan\).

Complementary angles and cofunction identities

Key idea

In any right triangle, the two acute angles add to \(90^\circ\). So if one acute angle is \(\theta\), the other is \(90^\circ-\theta\). This creates cofunction identities: \[ \sin(90^\circ-\theta)=\cos\theta,\quad \cos(90^\circ-\theta)=\sin\theta, \] \[ \tan(90^\circ-\theta)=\cot\theta,\quad \cot(90^\circ-\theta)=\tan\theta, \] \[ \sec(90^\circ-\theta)=\csc\theta,\quad \csc(90^\circ-\theta)=\sec\theta. \] A very common conversion is \(\tan(90^\circ-\theta)=\cot\theta=\dfrac{1}{\tan\theta}\).

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Summary

• Complementary angles: \(\theta\) and \(90^\circ-\theta\).
• Key identity: \(\tan(90^\circ-\theta)=\cot\theta=\dfrac{1}{\tan\theta}\).

Choosing the right ratio to solve a triangle

Key idea

To solve right-triangle problems efficiently, match what you know to the ratio that uses it:

• If you have opposite and hypotenuse, use \(\sin\theta\).
• If you have adjacent and hypotenuse, use \(\cos\theta\).
• If you have opposite and adjacent, use \(\tan\theta\).

If you need an angle from a ratio (like \(\cos\alpha=\frac{\sqrt3}{2}\)), use special angles (or inverse trig in later lessons).

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Summary

• Pick \(\sin\), \(\cos\), or \(\tan\) based on which sides you have.
• Use special triangles to recognize exact values quickly.

Why right-triangle ratios matter

Where right-triangle trigonometry shows up

• Geometry: triangle similarity, distances, and angle relationships.
• Science and engineering: component resolution, ramps, slopes, and navigation.
• Everyday problems: angles of elevation/depression, shadows, heights, and inclines.

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Final recap

• SOHCAHTOA: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\).
• Reciprocals: \(\csc=\dfrac{1}{\sin}\), \(\sec=\dfrac{1}{\cos}\), \(\cot=\dfrac{1}{\tan}\).
• Use the Pythagorean theorem to find missing sides before computing ratios.
• Special angles \(30^\circ,45^\circ,60^\circ\) come from \(30\text{-}60\text{-}90\) and \(45\text{-}45\text{-}90\) triangles.
• Complementary angles: \(\sin(90^\circ-\theta)=\cos\theta\), \(\tan(90^\circ-\theta)=\cot\theta\).