Without using a calculator, find the value of $\frac{\tan 80}{\cot 10}$.
Answer (1)
Use the identity cot x = tan ( 90 − x ) .
Substitute x = 10 into the identity to get cot 10 = tan ( 90 − 10 ) = tan 80 .
Substitute tan 80 for cot 10 in the original expression: t a n 80 t a n 80 .
Simplify the expression to get the final answer: 1 .
Explanation
Analyze the problem We are asked to find the value of the expression c o t 10 t a n 80 without using a calculator.
Use trigonometric identity We know that cot x = tan ( 9 0 ∘ − x ) . Let's use this identity to rewrite the denominator.
Calculate cotangent Substituting x = 1 0 ∘ into the identity, we get cot 1 0 ∘ = tan ( 9 0 ∘ − 1 0 ∘ ) = tan 8 0 ∘
Substitute in the original expression Now, substitute tan 8 0 ∘ for cot 1 0 ∘ in the original expression: cot 1 0 ∘ tan 8 0 ∘ = tan 8 0 ∘ tan 8 0 ∘
Simplify the expression Simplify the expression: tan 8 0 ∘ tan 8 0 ∘ = 1
State the final answer Therefore, the value of the expression c o t 10 t a n 80 is 1.
Examples
Imagine you're building a ramp for skateboarding and need to ensure a specific angle for optimal performance. If you know the tangent of one angle is equal to the cotangent of another (complementary) angle, you can easily adjust the ramp's design without complex calculations. This principle allows you to create structures with precise angles, ensuring functionality and safety in various applications, from sports equipment to architectural designs.
