Hyperbolic functions and inverse relations · 双曲函数及其反关系
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| hyperbolic cosine/ˌhaɪpəˈbɒlɪk ˈkəʊsaɪn/ | 双曲余弦 | shuāng qū yú xián |
How can growth and decay make a symmetric curve?
- A hanging cable has a curved profile related to exponentials. Hyperbolic functions combine growth and decay symmetrically.
- This lesson studies hyperbolic cosine 双曲余弦: The function cosh x=(e^x+e^(-x))/2.
Choose the mathematical structure
- Define sinh x=(e^x-e^(-x))/2 and cosh x=(e^x+e^(-x))/2. Their identity is cosh²x-sinh²x=1. Derivatives are sinh prime=cosh and cosh prime=sinh.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines hyperbolic cosine? · 哪种描述正确定义了双曲余弦？
The function cosh x=(e^x+e^(-x))/2. · 函数 cosh x=(e^x+e^(-x))/2。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
At x=ln2, e^x=2 and e^(-x)=1/2. Thus cosh x=1.25 and sinh x=0.75. Their squared difference is 1.5625-0.5625=1. To invert y=sinh x, solve a quadratic in e^x and choose the positive root.
Hyperbolic functions and inverse relations · 双曲函数及其反关系
Define sinh x=(e^x-e^(-x))/2 and cosh x=(e^x+e^(-x))/2 · 定义 sinh x=(e^x-e^(-x))/2 和 cosh x=(e^x+e^(-x))/2
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find cosh(ln2). · 求 cosh(ln2)。
e^(ln2)=2 and e^(-ln2)=1/2. Their half-sum is 1.25. · e^(ln2)=2 且 e^(-ln2)=1/2。它们的半和是 1.25。
Test a tempting shortcut
- The hyperbolic identity has a minus sign. cosh is not one-to-one on all real inputs; its usual inverse uses x≥0. Ordinary circular-trigonometric identities cannot be substituted unchanged.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The identity for hyperbolic functions is cosh²x+sinh²x=1. This claim is false. Explain which definition or assumption it violates.
Find sinh(ln2). · 求 sinh(ln2)。
Their half-difference is (2-1/2)/2=0.75. · 它们的半差为 (2-1/2)/2=0.75。
The identity for hyperbolic functions is cosh²x+sinh²x=1. · 双曲函数的恒等式为 cosh²x+sinh²x=1。
The hyperbolic identity has a minus sign. cosh is not one-to-one on all real inputs; its usual inverse uses x≥0. Ordinary circular-trigonometric identities cannot be substituted unchanged. · 双曲恒等式包含减号。cosh 在所有实数输入上并非单射；其通常的反函数使用 x≥0。普通的三角恒等式不能直接照搬不变地替换。
Interpret a new situation
- Use exponential definitions to prove identities and solve equations. State domain restrictions for inverse functions before differentiating or integrating them.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find cosh²(ln2)-sinh²(ln2). · 求 cosh²(ln2)-sinh²(ln2)。
1.25²-0.75²=1, agreeing with the hyperbolic identity. · 1.25²-0.75²=1，与双曲恒等式一致。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL pure mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The function cosh x=(e^x+e^(-x))/2. Choose the relationship, show the method, check its assumptions and interpret the result.