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🎯 Welcome to Inverse Trigonometry!

Let's explore special angles and their inverse trigonometric values step by step.

📚 What You'll Learn:

  • ✅ Special angles: 0°, 30°, 45°, 60°, 90°
  • ✅ Trigonometric values for these angles
  • ✅ Inverse trigonometry concepts
  • ✅ Interactive calculations
  • ✅ Practice exercises

🚀 Interactive Learning: You'll input numbers and see results on every slide!

Ready to start your journey? Click Next to begin!

📐 Understanding Special Angles

Special angles have exact trigonometric values. Let's explore them:

0° = \(0\) radians
30° = \(\frac{\pi}{6}\) radians
45° = \(\frac{\pi}{4}\) radians
60° = \(\frac{\pi}{3}\) radians
90° = \(\frac{\pi}{2}\) radians

🧮 Try Converting!

Convert degrees to radians:

📊 Trigonometric Values Table

Angle \(\sin\theta\) \(\cos\theta\) \(\tan\theta\)
0° \(0\) \(1\) \(0\)
30° \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{3}}{3}\)
45° \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) \(1\)
60° \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\)
90° \(1\) \(0\) undefined

🎯 Test Your Knowledge!

Enter an angle to see its trigonometric values:

🔄 What is Inverse Trigonometry?

Inverse trigonometric functions work backwards - they find the angle when given the trigonometric value.

\(\arcsin(x)\)

If \(\sin\theta = x\), then \(\theta = \arcsin(x)\)

Range: \([-90°, 90°]\)

Domain: \([-1, 1]\)

\(\arccos(x)\)

If \(\cos\theta = x\), then \(\theta = \arccos(x)\)

Range: \([0°, 180°]\)

Domain: \([-1, 1]\)

\(\arctan(x)\)

If \(\tan\theta = x\), then \(\theta = \arctan(x)\)

Range: \((-90°, 90°)\)

Domain: All real numbers

🧪 Try an Example!

Let's find: What angle has \(\sin\theta = \frac{1}{2}\)?

Hint: Look at the table from the previous slide!

📐 Interactive Arcsin Calculator

\(\arcsin(x)\) - Arcsine Function

The arcsine function finds the angle whose sine is the given value.

Key Points:

  • • Input must be between -1 and 1
  • • Output is between -90° and 90°
  • • \(\arcsin(\frac{1}{2}) = 30°\)
  • • \(\arcsin(\frac{\sqrt{2}}{2}) = 45°\)
  • • \(\arcsin(\frac{\sqrt{3}}{2}) = 60°\)

🧮 Try Different Values!

💡 Try These Values:

📐 Interactive Arccos Calculator

\(\arccos(x)\) - Arccosine Function

The arccosine function finds the angle whose cosine is the given value.

Key Points:

  • • Input must be between -1 and 1
  • • Output is between 0° and 180°
  • • \(\arccos(\frac{\sqrt{3}}{2}) = 30°\)
  • • \(\arccos(\frac{\sqrt{2}}{2}) = 45°\)
  • • \(\arccos(\frac{1}{2}) = 60°\)
  • • \(\arccos(0) = 90°\)

🧮 Try Different Values!

💡 Try These Values:

📐 Interactive Arctan Calculator

\(\arctan(x)\) - Arctangent Function

The arctangent function finds the angle whose tangent is the given value.

Key Points:

  • • Input can be any real number
  • • Output is between -90° and 90°
  • • \(\arctan(\frac{\sqrt{3}}{3}) = 30°\)
  • • \(\arctan(1) = 45°\)
  • • \(\arctan(\sqrt{3}) = 60°\)

🧮 Try Different Values!

💡 Try These Values:

🎯 Final Challenge Quiz

Question 1: Calculate \(\arcsin(\frac{1}{2})\)

Question 2: Calculate \(\arccos(\frac{\sqrt{2}}{2})\)

Question 3: Calculate \(\arctan(\sqrt{3})\)