Fill in the blanks to complete the fundamental identities.
$\tan \theta=\frac{1}{\square}=\frac{\sin \theta}{\square}$
Answer (1)
Recall the definition of tan θ in terms of sin θ and cos θ : tan θ = c o s θ s i n θ .
Recall the definition of tan θ in terms of cot θ : tan θ = c o t θ 1 .
Combine these identities to complete the given expression.
The complete identity is tan θ = c o t θ 1 = c o s θ s i n θ , so the answer is cot θ and cos θ .
Explanation
Understanding the Problem We are asked to fill in the blanks in the given trigonometric identity for tan θ . The given identity is tan θ = □ 1 = □ s i n θ . We need to find the expressions that complete the identity.
Recalling Trigonometric Identities We know that tan θ can be expressed in terms of sin θ and cos θ as: tan θ = cos θ sin θ Also, tan θ is the reciprocal of cot θ , so: tan θ = cot θ 1
Completing the Identity Comparing these identities with the given expression tan θ = □ 1 = □ s i n θ , we can fill in the blanks. The first blank should be cot θ and the second blank should be cos θ . Therefore, the complete identity is: tan θ = cot θ 1 = cos θ sin θ
Final Answer The completed fundamental identities are: tan θ = cot θ 1 = cos θ sin θ
Examples
Understanding trigonometric identities like this is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum, the tangent of the angle of displacement relates the pendulum's potential and kinetic energy. Similarly, in electrical engineering, the tangent function helps describe the phase relationship between voltage and current in AC circuits. These identities provide a foundation for solving complex problems in various scientific and technical domains.
