You are given the following information about θ
<span>sinθ=<span>23</span>,<span>π2</span><θ<π</span>
What are <span>cosθ</span> and <span>tanθ</span>?
Trigonometric Identities
You can use the Pythagorean, Tangent and Reciprocal Identities to find all six trigonometric values for certain angles. Let’s walk through a few problems so that you understand how to do this.
<span>Let's solve the following problems using trigonometric identities.</span>
<span>Given that <span>cosθ=<span>35</span></span> and <span>0<θ<<span>π2</span></span>, find <span>sinθ</span>.</span>
Use the Pythagorean Identity to find <span>sinθ</span>.
<span><span><span><span>sin2</span>θ+<span>cos2</span>θ</span><span><span>sin2</span>θ+<span><span>(<span>35</span>)</span>2</span></span><span><span>sin2</span>θ</span><span><span>sin2</span>θ</span><span>sinθ</span></span><span><span>=1</span><span>=1</span><span>=1−<span>925</span></span><span>=<span>1625</span></span><span>=±<span>45</span></span></span></span>
Because θ is in the first quadrant, we know that sine will be positive. <span>sinθ=<span>45</span></span>
<span>Find <span>tanθ</span> from #1 above.</span>
Use the Tangent Identity to find <span>tanθ</span>.
<span>tanθ=<span><span>sinθ</span><span>cosθ</span></span>=<span><span>45</span><span>35</span></span>=<span>43</span></span>
<span>Find the other three trigonometric functions of θ from #1.</span>
To find secant, cosecant, and cotangent use the Reciprocal Identities.
<span>cscθ=<span>1<span>sinθ</span></span>=<span>1<span>45</span></span>=<span>54</span>secθ=<span>1<span>cosθ</span></span>=<span>1<span>35</span></span>=<span>53</span>cotθ=<span>1<span>tanθ</span></span>=<span>1<span>43</span></span>=<span>34</span></span>
ExamplesExample 1
Earlier, you were asked to find <span>cosθ</span> and <span>tanθ</span> of <span>sinθ=<span>23</span>,<span>π2</span><θ<π</span>.
First, use the Pythagorean Identity to find <span>cosθ</span>.
<span><span><span><span>sin2</span>θ+<span>cos2</span>θ</span><span>(<span>23</span><span>)2</span>+<span>cos2</span>θ=1</span><span><span>cos2</span>θ</span><span><span>cos2</span>θ</span><span>cosθ</span></span><span><span>=1</span><span>=1−<span>49</span></span><span>=<span>59</span></span><span>=±<span><span>5√</span>3</span></span></span></span>
However, because θ is restricted to the second quadrant, the cosine must be negative. Therefore, <span>cosθ=−<span><span>5√</span>3</span></span>.
Now use the Tangent Identity to find <span>tanθ</span>.
<span>tanθ=<span><span>sinθ</span><span>cosθ</span></span>=<span><span>23</span><span>−<span><span>5√</span>3</span></span></span>=<span><span>−2</span><span>5√</span></span>=<span><span>−2<span>5√</span></span>5</span></span>
<span>Find the values of the other five trigonometric functions.</span>
Example 2
<span>tanθ=−<span>512</span>,<span>π2</span><θ<π</span>
First, we know that θ is in the second quadrant, making sine positive and cosine negative. For this problem, we will use the Pythagorean Identity <span>1+<span>tan2</span>θ=<span>sec2</span>θ</span> to find secant.
<span><span><span>1+<span><span>(−<span>512</span>)</span>2</span></span><span>1+<span>25144</span></span><span>169144</span><span>±<span>1312</span></span><span>−<span>1312</span></span></span><span><span>=<span>sec2</span>θ</span><span>=<span>sec2</span>θ</span><span>=<span>sec2</span>θ</span><span>=secθ</span><span>=secθ</span></span></span>
If <span>secθ=−<span>1312</span></span>, then <span>cosθ=−<span>1213</span></span>. <span>sinθ=<span>513</span></span> because the numerator value of tangent is the sine and it has the same denominator value as cosine. <span>cscθ=<span>135</span></span> and <span>cotθ=−<span>125</span></span> from the Reciprocal Identities.
Example 3
<span>cscθ=−8,π<θ<<span><span>3π</span>2</span></span>
θ is in the third quadrant, so both sine and cosine are negative. The reciprocal of <span>cscθ=−8</span>, will give us <span>sinθ=−<span>18</span></span>. Now, use the Pythagorean Identity <span><span>sin2</span>θ+<span>cos2</span>θ=1</span> to find cosine.
<span><span><span><span><span>(−<span>18</span>)</span>2</span>+<span>cos2</span>θ</span><span><span>cos2</span>θ</span><span><span>cos2</span>θ</span><span>cosθ</span><span>cosθ</span></span><span><span>=1</span><span>=1−<span>164</span></span><span>=<span>6364</span></span><span>=±<span><span>3<span>7√</span></span>8</span></span><span>=−<span><span>3<span>7√</span></span>8</span></span></span></span>
<span>secθ=−<span>8<span>3<span>7√</span></span></span>=−<span><span>8<span>7√</span></span>21</span>,tanθ=<span>1<span>3<span>7√</span></span></span>=<span><span>7√</span>21</span>,</span> and <span>cotθ=3<span>7√</span></span>
Review<span>In which quadrants is the sine value positive? Negative?In which quadrants is the cosine value positive? Negative?In which quadrants is the tangent value positive? Negative?</span>
Find the values of the other five trigonometric functions of θ.
<span><span>sinθ=<span>817</span>,0<θ<<span>π2</span></span><span>cosθ=−<span>56</span>,<span>π2</span><θ<π</span><span>tanθ=<span><span>3√</span>4</span>,0<θ<<span>π2</span></span><span>secθ=−<span>419</span>,π<θ<<span><span>3π</span>2</span></span><span>sinθ=−<span>1114</span>,<span><span>3π</span>2</span><θ<2π</span><span>cosθ=<span><span>2√</span>2</span>,0<θ<<span>π2</span></span><span>cotθ=<span>5√</span>,π<θ<<span><span>3π</span>2</span></span><span>cscθ=4,<span>π2</span><θ<π</span><span>tanθ=−<span>710</span>,<span><span>3π</span>2</span><θ<2π</span>Aside from using the identities, how else can you find the values of the other five trigonometric functions?<span>Given that <span>cosθ=<span>611</span></span> and θ is in the <span>2<span>nd</span></span> quadrant, what is <span>sin(−θ)</span>?</span><span>Given that <span>tanθ=−<span>58</span></span> and θ is in the <span>4<span>th</span></span> quadrant, what is <span>sec(−θ)</span>?</span></span>