Functions
>
Trigonometric, Log, and Exponential
> Example: Trigonometric Identities
Example: Trigonometric Identities
Basic Identities
<region id="ID0EYQ5DB" top="153.60000000000002" left="19.200000000000003" height="47.2" width="117.60333333333335" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="48"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <div /> <apply> <id xml:space="preserve">sin</id> <id xml:space="preserve">x</id> </apply> <apply> <id xml:space="preserve">cos</id> <id xml:space="preserve">x</id> </apply> </apply> <apply> <id xml:space="preserve">tan</id> <id xml:space="preserve">x</id> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EGR5DB" top="211.20000000000005" left="19.200000000000003" height="34" width="150.05666666666664" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="50"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <plus /> <apply> <pow /> <apply> <id xml:space="preserve">sin</id> <id xml:space="preserve">x</id> </apply> <real>2</real> </apply> <apply> <pow /> <apply> <id xml:space="preserve">cos</id> <id xml:space="preserve">x</id> </apply> <real>2</real> </apply> </apply> <real>1</real> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EUR5DB" top="249.60000000000002" left="19.200000000000003" height="47.2" width="114.91666666666669" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="52"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">sec</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">cos</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0ECS5DB" top="288.00000000000006" left="19.200000000000003" height="47.2" width="114.19666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="54"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">csc</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">sin</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EQS5DB" top="326.40000000000003" left="19.200000000000003" height="47.2" width="117.12" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="56"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">cot</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">tan</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
Complex Identities
<region id="ID0EHT5DB" top="441.60000000000008" left="19.200000000000003" height="49.352000000000004" width="137.53000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="58"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">sin</id> <id xml:space="preserve">z</id> </apply> <apply> <div /> <apply> <minus /> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <mult /> <apply> <neg /> <id xml:space="preserve">i</id> </apply> <id xml:space="preserve">z</id> </apply> </apply> </apply> <apply> <mult /> <real>2</real> <id xml:space="preserve">i</id> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EVT5DB" top="480.00000000000006" left="19.200000000000003" height="49.352000000000004" width="137.83666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="60"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">cos</id> <id xml:space="preserve">z</id> </apply> <apply> <div /> <apply> <plus /> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <mult /> <apply> <neg /> <id xml:space="preserve">i</id> </apply> <id xml:space="preserve">z</id> </apply> </apply> </apply> <real>2</real> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EDU5DB" top="537.60000000000014" left="19.200000000000003" height="27.752000000000002" width="158.92000000000002" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="62"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> <apply> <plus /> <apply> <id xml:space="preserve">cos</id> <id xml:space="preserve">z</id> </apply> <apply> <mult /> <id xml:space="preserve">i</id> <apply> <id xml:space="preserve">sin</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0ERU5DB" top="576.00000000000011" left="19.200000000000003" height="27.752000000000002" width="215.66666666666669" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="64"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <plus /> <id xml:space="preserve">x</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">y</id> </apply> </apply> </apply> <apply> <mult /> <apply> <pow /> <id xml:space="preserve">e</id> <id xml:space="preserve">x</id> </apply> <parens> <apply> <plus /> <apply> <id xml:space="preserve">cos</id> <id xml:space="preserve">y</id> </apply> <apply> <mult /> <id xml:space="preserve">i</id> <apply> <id xml:space="preserve">sin</id> <id xml:space="preserve">y</id> </apply> </apply> </apply> </parens> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E6U5DB" top="633.60000000000014" left="19.200000000000003" height="47.2" width="279.2166666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="66"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">ln</id> <apply> <plus /> <id xml:space="preserve">x</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">y</id> </apply> </apply> </apply> <apply> <plus /> <apply> <scale /> <apply> <div /> <real>1</real> <real>2</real> </apply> <apply> <id xml:space="preserve">ln</id> <apply> <plus /> <apply> <pow /> <id xml:space="preserve">x</id> <real>2</real> </apply> <apply> <pow /> <id xml:space="preserve">y</id> <real>2</real> </apply> </apply> </apply> </apply> <apply> <mult /> <id xml:space="preserve">i</id> <apply> <id xml:space="preserve">atan</id> <apply> <div /> <id xml:space="preserve">y</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
The
ln0
function is similar to
ln
, but it is defined for zero.
<region id="ID0EMX5DB" top="729.60000000000014" left="38.400000000000006" height="25.6" width="76.166666666666671" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="68"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <id xml:space="preserve">ln</id> <real>0</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> </math> </region>
<region id="ID0E1X5DB" top="748.80000000000007" left="38.400000000000006" height="27.752000000000002" width="135.35333333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="70"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">ln0</id> <real>0</real> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <general precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" /> </resultFormat> </math> </region>
Hyperbolic Functions and Exponentials
<region id="ID0ESY5DB" top="844.80000000000018" left="38.400000000000006" height="49.352000000000004" width="131.08333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="72"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">sinh</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <apply> <minus /> <apply> <pow /> <id xml:space="preserve">e</id> <id xml:space="preserve">x</id> </apply> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <neg /> <id xml:space="preserve">x</id> </apply> </apply> </apply> <real>2</real> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EAZ5DB" top="883.20000000000016" left="38.400000000000006" height="49.352000000000004" width="131.39000000000002" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="74"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">cosh</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <apply> <plus /> <apply> <pow /> <id xml:space="preserve">e</id> <id xml:space="preserve">x</id> </apply> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <neg /> <id xml:space="preserve">x</id> </apply> </apply> </apply> <real>2</real> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EOZ5DB" top="940.80000000000018" left="38.400000000000006" height="51.504000000000005" width="133.66666666666666" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="76"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">tanh</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <apply> <minus /> <apply> <pow /> <id xml:space="preserve">e</id> <id xml:space="preserve">x</id> </apply> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <neg /> <id xml:space="preserve">x</id> </apply> </apply> </apply> <apply> <plus /> <apply> <pow /> <id xml:space="preserve">e</id> <id xml:space="preserve">x</id> </apply> <apply> <pow /> <id xml:space="preserve">e</id> <apply> <neg /> <id xml:space="preserve">x</id> </apply> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E3Z5DB" top="979.20000000000016" left="38.400000000000006" height="47.2" width="131.33" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="78"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">csch</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">sinh</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EK15DB" top="1036.8000000000002" left="38.400000000000006" height="47.2" width="132.05" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="80"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">sech</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">cosh</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EY15DB" top="1075.2000000000003" left="38.400000000000006" height="47.2" width="134.25333333333333" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="82"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">coth</id> <id xml:space="preserve">x</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">tanh</id> <id xml:space="preserve">x</id> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
Hyperbolic Functions and Trigonometric Relationships
<region id="ID0EP25DB" top="1190.4" left="19.200000000000003" height="25.6" width="154.07" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="84"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">sinh</id> <id xml:space="preserve">z</id> </apply> <apply> <mult /> <apply> <neg /> <id xml:space="preserve">i</id> </apply> <apply> <id xml:space="preserve">sin</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E425DB" top="1248.0000000000002" left="19.200000000000003" height="25.6" width="129.99" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="86"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">cosh</id> <id xml:space="preserve">z</id> </apply> <apply> <id xml:space="preserve">cos</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EL35DB" top="1286.4" left="19.200000000000003" height="25.6" width="159.23666666666668" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="88"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">tanh</id> <id xml:space="preserve">z</id> </apply> <apply> <mult /> <apply> <neg /> <id xml:space="preserve">i</id> </apply> <apply> <id xml:space="preserve">tan</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EZ35DB" top="1344.0000000000002" left="19.200000000000003" height="47.2" width="133.66000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="90"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">csch</id> <id xml:space="preserve">z</id> </apply> <apply> <div /> <id xml:space="preserve">i</id> <apply> <id xml:space="preserve">sin</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EH45DB" top="1382.4" left="19.200000000000003" height="47.2" width="134.38" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="92"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">sech</id> <id xml:space="preserve">z</id> </apply> <apply> <div /> <real>1</real> <apply> <id xml:space="preserve">cos</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EV45DB" top="1440.0000000000002" left="19.200000000000003" height="47.2" width="136.58333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="94"> <apply xmlns="http://schemas.mathsoft.com/math50"> <equal /> <apply> <id xml:space="preserve">coth</id> <id xml:space="preserve">z</id> </apply> <apply> <div /> <id xml:space="preserve">i</id> <apply> <id xml:space="preserve">tan</id> <apply> <mult /> <id xml:space="preserve">i</id> <id xml:space="preserve">z</id> </apply> </apply> </apply> </apply> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
Copy Expressions
Related TopicsĀ
Hyperbolic Functions
Trigonometric Functions
Was this helpful?
This site works best with JavaScript enabled