How to Use 30-60-90 Triangles in Trigonometry

TL;DR

30-60-90 triangles have side lengths of 1, √3, and 2. The sine, cosine, tangent, cosecant, secant, and cotangent values for 30 degrees are 1/2, √3/2, √3/3, 2, 2√3/3, and √3, respectively. For 60 degrees, these values are √3/2, 2/√3, √3, and 1/2. Utilizing these triangles simplifies angle evaluation without memorizing the unit circle.

Transcript

the next special triangle that we need to talk about is the 30-60-90 triangle across the 30 is 1 across the 60 is 1 times root 3 or the square root of 3. across the 90 is 2. so let's say if we wish to evaluate sine 30 opposite to 30 is 1. the hypotenuse is going to be 2. so sine theta is opposite over hypotenuse so it's 1 over 2. now if we wish to ... Read More

Key Insights

• 🔺 The 30-60-90 triangle has side lengths 1, √3, and 2.
• 🔺 The values of sine, cosine, tangent, cosecant, secant, and cotangent for angles 30 and 60 in the 30-60-90 triangle can be determined using the corresponding sides.
• 🔺 The 45-45-90 triangle has side lengths 1, 1, and √2.
• 🔺 The values of sine, cosine, tangent, cosecant, secant, and cotangent for angle 45 in the 45-45-90 triangle can be found using the appropriate side ratios.
• 🔺 The special right triangles eliminate the need to memorize the unit circle for evaluating common angles.
• ❓ Sine 30 is 1/2, cosecant 30 is 2, cosine 30 is √3/2, secant 30 is 2√3/3, tangent 30 is √3/3, and cotangent 30 is √3.

Questions & Answers

Q: What are the side lengths of a 30-60-90 triangle?

The side lengths of a 30-60-90 triangle are 1, √3, and 2.

Q: What is the value of sine 30 in a 30-60-90 triangle?

The value of sine 30 in a 30-60-90 triangle is 1/2.

Q: How do you find secant 60 in a 45-45-90 triangle?

In a 45-45-90 triangle, secant 60 can be found as the reciprocal of cosine 60, which is 2.

Q: What is the value of tangent 45 in a 45-45-90 triangle?

The value of tangent 45 in a 45-45-90 triangle is 1.

Summary & Key Takeaways

• The 30-60-90 triangle has side lengths 1, √3, and 2.

• Sine 30 is 1/2, cosecant 30 is 2, cosine 30 is √3/2, secant 30 is 2√3/3, tangent 30 is √3/3, and cotangent 30 is √3.

• The 45-45-90 triangle has side lengths 1, 1, and √2.

• Sine 45 is 1/√2, cosecant 45 is √2, cosine 45 is 1/√2, secant 45 is √2, tangent 45 is 1, and cotangent 45 is 1.

• The 30-60-90 and 45-45-90 triangles can be used to evaluate common angles without the need to memorize the unit circle.