How to Write in Algebraic Form csc(arctan(x/sqrt(2))

TL;DR

Simplifying trigonometric functions into algebraic form using inverse tangent and Pythagorean theorem.

Transcript

so the question is asking us to write this in what's called algebraic form that basically means we have to get rid of all of the trig functions and end up with purely functions of X so let's go ahead and work through it carefully so the first step in these problems is always to take this piece here and call that piece theta so step one we're going ... Read More

Key Insights

• ❓ Initiate simplification by assigning theta and using inverse tangent.
• 😑 Utilize SOHCAHTOA to draw and solve the triangle formed by the trigonometric expression.
• 🔺 Apply the Pythagorean theorem to find the missing side of the triangle.

Questions & Answers

Q: How is the process of simplifying trigonometric functions into algebraic form initiated?

The process begins by assigning a variable, theta, to the inner trigonometric expression and applying the inverse tangent function, linking x over the square root of two to theta.

Q: What role does SOHCAHTOA play in simplifying trigonometric functions?

SOHCAHTOA is utilized to draw a triangle with the opposite side as x, adjacent side as the square root of two, and hypotenuse, facilitating the simplification process by defining the relationships between the sides.

Q: How is the Pythagorean theorem incorporated into the simplification of trigonometric functions?

The Pythagorean theorem is utilized to find the missing side of the triangle, representing the hypotenuse, by taking the square root of the sum of the squares of the other two sides.

Q: What is the final step in simplifying trigonometric functions into algebraic form?

The final step involves going back to the original trigonometric function, such as cosecant of theta, and applying the principles of SOHCAHTOA to simplify it into a purely algebraic expression.

Summary & Key Takeaways

• Simplifying trigonometric functions involves assigning a variable, obtaining the inverse function, applying SOHCAHTOA, drawing a triangle, and solving using the Pythagorean theorem.

• Inverse tangent is used to relate x over the square root of two to theta, after which the tangent function relates theta back to x over the square root of two.

• The process involves drawing a triangle with the sides x, square root of two, and the hypotenuse (solved using the Pythagorean theorem) to simplify trigonometric functions.